\(sl(2)\)-subalgebras of sp(16), type \(C^{1}_8\)

sp(16), type \(C^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

Number of sl(2) subalgebras: 99.

Let h be in the Cartan subalgebra. Let \(\alpha_1, ..., \alpha_n\) be simple roots with respect to h. Then the h-characteristic, as defined by E. Dynkin, is the n-tuple \((\alpha_1(h), ..., \alpha_n(h))\).

The actual realization of h. The coordinates of h are given with respect to the fixed original simple basis. Note that the h-characteristic is computed using a possibly different simple basis, more precisely, with respect to any h-positive simple basis.

A regular semisimple subalgebra might contain an sl(2) such that the sl(2) has no centralizer in the regular semisimple subalgebra, but the regular semisimple subalgebra might fail to be minimal containing. This happens when another minimal containing regular semisimple subalgebra of equal rank nests as a root subalgebra in the containing SA. See Dynkin, Semisimple Lie subalgebras of semisimple Lie algebras, remark before Theorem 10.4.

The \(sl(2)\) submodules of the ambient Lie algebra are parametrized by their highest weight with respect to the Cartan element h of \(sl(2)\). In turn, the highest weight is a positive integer multiple of the fundamental highest weight \(\psi\). \(V_{l\psi}\) is \(l + 1\)-dimensional.

Type + realization link	h-Characteristic	Realization of h	sl(2)-module decomposition of the ambient Lie algebra \(\psi=\) the fundamental \(sl(2)\)-weight.	Centralizer dimension	Type of semisimple part of centralizer, if known	The square of the length of the weight dual to h.	Dynkin index	Minimal containing regular semisimple SAs	Containing regular semisimple SAs in which the sl(2) has no centralizer
\(A^{680}_1\)	(2, 2, 2, 2, 2, 2, 2, 2)	(30, 56, 78, 96, 110, 120, 126, 64)	\(V_{30\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}\)	0	\(\displaystyle 0\)	1360	680	C^{1}_8;	C^{1}_8;
\(A^{456}_1\)	(2, 2, 2, 2, 2, 2, 0, 2)	(26, 48, 66, 80, 90, 96, 98, 50)	\(V_{26\psi}+V_{22\psi}+V_{18\psi}+2V_{14\psi}+V_{12\psi}+V_{10\psi}+V_{6\psi}+2V_{2\psi}\)	0	\(\displaystyle 0\)	912	456	C^{1}_8; C^{1}_7+A^{1}_1;	C^{1}_8; C^{1}_7+A^{1}_1;
\(A^{455}_1\)	(2, 2, 2, 2, 2, 2, 1, 0)	(26, 48, 66, 80, 90, 96, 98, 49)	\(V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+2V_{13\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	910	455	C^{1}_7;	C^{1}_7;
\(A^{296}_1\)	(2, 2, 2, 2, 0, 2, 0, 2)	(22, 40, 54, 64, 70, 76, 78, 40)	\(V_{22\psi}+V_{18\psi}+2V_{14\psi}+V_{12\psi}+2V_{10\psi}+V_{8\psi}+2V_{6\psi}+2V_{2\psi}\)	0	\(\displaystyle 0\)	592	296	C^{1}_8; C^{1}_6+B^{1}_2;	C^{1}_8; C^{1}_6+B^{1}_2;
\(A^{288}_1\)	(2, 2, 2, 2, 2, 0, 0, 2)	(22, 40, 54, 64, 70, 72, 74, 38)	\(V_{22\psi}+V_{18\psi}+V_{14\psi}+2V_{12\psi}+3V_{10\psi}+V_{6\psi}+4V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	576	288	C^{1}_7+A^{1}_1; C^{1}_6+2A^{1}_1; C^{1}_6+A^{2}_1;	C^{1}_7+A^{1}_1; C^{1}_6+2A^{1}_1; C^{1}_6+A^{2}_1;
\(A^{287}_1\)	(2, 2, 2, 2, 2, 0, 1, 0)	(22, 40, 54, 64, 70, 72, 74, 37)	\(V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{11\psi}+2V_{10\psi}+V_{6\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	574	287	C^{1}_7; C^{1}_6+A^{1}_1;	C^{1}_7; C^{1}_6+A^{1}_1;
\(A^{286}_1\)	(2, 2, 2, 2, 2, 1, 0, 0)	(22, 40, 54, 64, 70, 72, 72, 36)	\(V_{22\psi}+V_{18\psi}+V_{14\psi}+4V_{11\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}+10V_{0}\)	10	\(\displaystyle B^{1}_2\)	572	286	C^{1}_6;	C^{1}_6;
\(A^{200}_1\)	(2, 2, 0, 2, 0, 2, 0, 2)	(18, 32, 42, 52, 58, 64, 66, 34)	\(V_{18\psi}+2V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+2V_{2\psi}\)	0	\(\displaystyle 0\)	400	200	C^{1}_8; C^{1}_5+C^{1}_3;	C^{1}_8; C^{1}_5+C^{1}_3;
\(A^{176}_1\)	(2, 2, 2, 0, 2, 0, 0, 2)	(18, 32, 42, 48, 54, 56, 58, 30)	\(V_{18\psi}+V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{8\psi}+3V_{6\psi}+V_{4\psi}+4V_{2\psi}\)	0	\(\displaystyle 0\)	352	176	C^{1}_8; C^{1}_7+A^{1}_1; C^{1}_6+B^{1}_2; C^{1}_5+C^{1}_3; C^{1}_5+B^{1}_2+A^{1}_1;	C^{1}_8; C^{1}_7+A^{1}_1; C^{1}_6+B^{1}_2; C^{1}_5+C^{1}_3; C^{1}_5+B^{1}_2+A^{1}_1;
\(A^{175}_1\)	(2, 2, 2, 0, 2, 0, 1, 0)	(18, 32, 42, 48, 54, 56, 58, 29)	\(V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}+3V_{6\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	350	175	C^{1}_7; C^{1}_5+B^{1}_2;	C^{1}_7; C^{1}_5+B^{1}_2;
\(A^{173}_1\)	(2, 2, 2, 1, 0, 1, 1, 0)	(18, 32, 42, 48, 52, 56, 58, 29)	\(V_{18\psi}+V_{14\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+2V_{7\psi}+V_{6\psi}+3V_{4\psi}+2V_{2\psi}+3V_{0}\)	3	not computed	346	173	C^{1}_5+A^{2}_2;	C^{1}_5+A^{2}_2;
\(A^{168}_1\)	(2, 2, 2, 2, 0, 0, 0, 2)	(18, 32, 42, 48, 50, 52, 54, 28)	\(V_{18\psi}+V_{14\psi}+4V_{10\psi}+3V_{8\psi}+V_{6\psi}+7V_{2\psi}+3V_{0}\)	3	not computed	336	168	C^{1}_6+2A^{1}_1; C^{1}_5+3A^{1}_1; C^{1}_6+A^{2}_1; C^{1}_5+A^{2}_1+A^{1}_1;	C^{1}_6+2A^{1}_1; C^{1}_5+3A^{1}_1; C^{1}_6+A^{2}_1; C^{1}_5+A^{2}_1+A^{1}_1;
\(A^{168}_1\)	(0, 2, 0, 2, 0, 2, 0, 2)	(14, 28, 38, 48, 54, 60, 62, 32)	\(3V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+3V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	336	168	2C^{1}_4; A^{2}_7;	2C^{1}_4; A^{2}_7;
\(A^{167}_1\)	(2, 2, 2, 2, 0, 0, 1, 0)	(18, 32, 42, 48, 50, 52, 54, 27)	\(V_{18\psi}+V_{14\psi}+3V_{10\psi}+2V_{9\psi}+2V_{8\psi}+V_{6\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}\)	4	\(\displaystyle A^{1}_1\)	334	167	C^{1}_6+A^{1}_1; C^{1}_5+2A^{1}_1; C^{1}_5+A^{2}_1;	C^{1}_6+A^{1}_1; C^{1}_5+2A^{1}_1; C^{1}_5+A^{2}_1;
\(A^{166}_1\)	(2, 2, 2, 2, 0, 1, 0, 0)	(18, 32, 42, 48, 50, 52, 52, 26)	\(V_{18\psi}+V_{14\psi}+2V_{10\psi}+4V_{9\psi}+V_{8\psi}+V_{6\psi}+2V_{2\psi}+4V_{\psi}+10V_{0}\)	10	\(\displaystyle B^{1}_2\)	332	166	C^{1}_6; C^{1}_5+A^{1}_1;	C^{1}_6; C^{1}_5+A^{1}_1;
\(A^{165}_1\)	(2, 2, 2, 2, 1, 0, 0, 0)	(18, 32, 42, 48, 50, 50, 50, 25)	\(V_{18\psi}+V_{14\psi}+V_{10\psi}+6V_{9\psi}+V_{6\psi}+V_{2\psi}+21V_{0}\)	21	\(\displaystyle C^{1}_3\)	330	165	C^{1}_5;	C^{1}_5;
\(A^{120}_1\)	(2, 0, 2, 0, 2, 0, 0, 2)	(14, 24, 34, 40, 46, 48, 50, 26)	\(V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{8\psi}+5V_{6\psi}+2V_{4\psi}+4V_{2\psi}\)	0	\(\displaystyle 0\)	240	120	C^{1}_8; C^{1}_7+A^{1}_1; C^{1}_5+C^{1}_3; 2C^{1}_4; C^{1}_4+C^{1}_3+A^{1}_1;	C^{1}_8; C^{1}_7+A^{1}_1; C^{1}_5+C^{1}_3; 2C^{1}_4; 2C^{1}_4; C^{1}_4+C^{1}_3+A^{1}_1;
\(A^{119}_1\)	(2, 0, 2, 0, 2, 0, 1, 0)	(14, 24, 34, 40, 46, 48, 50, 25)	\(V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+V_{4\psi}+3V_{2\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	238	119	C^{1}_7; C^{1}_4+C^{1}_3;	C^{1}_7; C^{1}_4+C^{1}_3;
\(A^{113}_1\)	(0, 2, 0, 2, 0, 1, 1, 0)	(12, 24, 32, 40, 44, 48, 50, 25)	\(3V_{12\psi}+V_{10\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{2\psi}+3V_{0}\)	3	not computed	226	113	A^{2}_6+A^{1}_1;	A^{2}_6+A^{1}_1;
\(A^{112}_1\)	(0, 2, 0, 2, 0, 2, 0, 0)	(12, 24, 32, 40, 44, 48, 48, 24)	\(3V_{12\psi}+V_{10\psi}+3V_{8\psi}+5V_{6\psi}+3V_{4\psi}+V_{2\psi}+6V_{0}\)	6	not computed	224	112	A^{2}_6;	A^{2}_6;
\(A^{104}_1\)	(2, 2, 0, 0, 2, 0, 0, 2)	(14, 24, 30, 36, 42, 44, 46, 24)	\(V_{14\psi}+3V_{10\psi}+2V_{8\psi}+6V_{6\psi}+3V_{4\psi}+4V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	208	104	C^{1}_6+B^{1}_2; C^{1}_4+2B^{1}_2; C^{1}_4+A^{2}_3;	C^{1}_6+B^{1}_2; C^{1}_4+2B^{1}_2; C^{1}_4+A^{2}_3;
\(A^{96}_1\)	(2, 2, 0, 2, 0, 0, 0, 2)	(14, 24, 30, 36, 38, 40, 42, 22)	\(V_{14\psi}+2V_{10\psi}+3V_{8\psi}+5V_{6\psi}+3V_{4\psi}+7V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	192	96	C^{1}_7+A^{1}_1; C^{1}_6+2A^{1}_1; C^{1}_5+C^{1}_3; C^{1}_5+B^{1}_2+A^{1}_1; C^{1}_4+C^{1}_3+A^{1}_1; C^{1}_4+B^{1}_2+2A^{1}_1; C^{1}_6+A^{2}_1; C^{1}_4+B^{1}_2+A^{2}_1;	C^{1}_7+A^{1}_1; C^{1}_6+2A^{1}_1; C^{1}_5+C^{1}_3; C^{1}_5+B^{1}_2+A^{1}_1; C^{1}_4+C^{1}_3+A^{1}_1; C^{1}_4+B^{1}_2+2A^{1}_1; C^{1}_6+A^{2}_1; C^{1}_4+B^{1}_2+A^{2}_1;
\(A^{95}_1\)	(2, 2, 0, 2, 0, 0, 1, 0)	(14, 24, 30, 36, 38, 40, 42, 21)	\(V_{14\psi}+2V_{10\psi}+2V_{8\psi}+2V_{7\psi}+4V_{6\psi}+2V_{4\psi}+2V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	190	95	C^{1}_7; C^{1}_6+A^{1}_1; C^{1}_5+B^{1}_2; C^{1}_4+C^{1}_3; C^{1}_4+B^{1}_2+A^{1}_1;	C^{1}_7; C^{1}_6+A^{1}_1; C^{1}_5+B^{1}_2; C^{1}_4+C^{1}_3; C^{1}_4+B^{1}_2+A^{1}_1;
\(A^{94}_1\)	(2, 2, 0, 2, 0, 1, 0, 0)	(14, 24, 30, 36, 38, 40, 40, 20)	\(V_{14\psi}+2V_{10\psi}+V_{8\psi}+4V_{7\psi}+3V_{6\psi}+V_{4\psi}+4V_{3\psi}+2V_{2\psi}+10V_{0}\)	10	\(\displaystyle B^{1}_2\)	188	94	C^{1}_6; C^{1}_4+B^{1}_2;	C^{1}_6; C^{1}_4+B^{1}_2;
\(A^{93}_1\)	(2, 2, 1, 0, 1, 0, 1, 0)	(14, 24, 30, 34, 38, 40, 42, 21)	\(V_{14\psi}+V_{10\psi}+2V_{9\psi}+V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+3V_{2\psi} +2V_{\psi}+3V_{0}\)	3	not computed	186	93	C^{1}_5+A^{2}_2; C^{1}_4+A^{2}_2+A^{1}_1;	C^{1}_5+A^{2}_2; C^{1}_4+A^{2}_2+A^{1}_1;
\(A^{92}_1\)	(2, 2, 1, 0, 1, 1, 0, 0)	(14, 24, 30, 34, 38, 40, 40, 20)	\(V_{14\psi}+V_{10\psi}+2V_{9\psi}+4V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+6V_{2\psi}+6V_{0}\)	6	not computed	184	92	C^{1}_4+A^{2}_2;	C^{1}_4+A^{2}_2;
\(A^{88}_1\)	(2, 2, 2, 0, 0, 0, 0, 2)	(14, 24, 30, 32, 34, 36, 38, 20)	\(V_{14\psi}+V_{10\psi}+4V_{8\psi}+5V_{6\psi}+11V_{2\psi}+6V_{0}\)	6	not computed	176	88	C^{1}_5+3A^{1}_1; C^{1}_4+4A^{1}_1; C^{1}_5+A^{2}_1+A^{1}_1; C^{1}_4+A^{2}_1+2A^{1}_1; C^{1}_4+2A^{2}_1;	C^{1}_5+3A^{1}_1; C^{1}_4+4A^{1}_1; C^{1}_5+A^{2}_1+A^{1}_1; C^{1}_4+A^{2}_1+2A^{1}_1; C^{1}_4+2A^{2}_1;
\(A^{87}_1\)	(2, 2, 2, 0, 0, 0, 1, 0)	(14, 24, 30, 32, 34, 36, 38, 19)	\(V_{14\psi}+V_{10\psi}+3V_{8\psi}+2V_{7\psi}+4V_{6\psi}+7V_{2\psi}+6V_{\psi}+6V_{0}\)	6	not computed	174	87	C^{1}_5+2A^{1}_1; C^{1}_4+3A^{1}_1; C^{1}_5+A^{2}_1; C^{1}_4+A^{2}_1+A^{1}_1;	C^{1}_5+2A^{1}_1; C^{1}_4+3A^{1}_1; C^{1}_5+A^{2}_1; C^{1}_4+A^{2}_1+A^{1}_1;
\(A^{86}_1\)	(2, 2, 2, 0, 0, 1, 0, 0)	(14, 24, 30, 32, 34, 36, 36, 18)	\(V_{14\psi}+V_{10\psi}+2V_{8\psi}+4V_{7\psi}+3V_{6\psi}+4V_{2\psi}+8V_{\psi}+11V_{0}\)	11	\(\displaystyle B^{1}_2\)	172	86	C^{1}_5+A^{1}_1; C^{1}_4+2A^{1}_1; C^{1}_4+A^{2}_1;	C^{1}_5+A^{1}_1; C^{1}_4+2A^{1}_1; C^{1}_4+A^{2}_1;
\(A^{85}_1\)	(2, 2, 2, 0, 1, 0, 0, 0)	(14, 24, 30, 32, 34, 34, 34, 17)	\(V_{14\psi}+V_{10\psi}+V_{8\psi}+6V_{7\psi}+2V_{6\psi}+2V_{2\psi}+6V_{\psi}+21V_{0}\)	21	\(\displaystyle C^{1}_3\)	170	85	C^{1}_5; C^{1}_4+A^{1}_1;	C^{1}_5; C^{1}_4+A^{1}_1;
\(A^{84}_1\)	(2, 2, 2, 1, 0, 0, 0, 0)	(14, 24, 30, 32, 32, 32, 32, 16)	\(V_{14\psi}+V_{10\psi}+8V_{7\psi}+V_{6\psi}+V_{2\psi}+36V_{0}\)	36	\(\displaystyle C^{1}_4\)	168	84	C^{1}_4;	C^{1}_4;
\(A^{80}_1\)	(0, 2, 0, 0, 2, 0, 0, 2)	(10, 20, 26, 32, 38, 40, 42, 22)	\(3V_{10\psi}+3V_{8\psi}+6V_{6\psi}+3V_{4\psi}+6V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	160	80	C^{1}_5+C^{1}_3; 2C^{1}_3+B^{1}_2; A^{2}_5+B^{1}_2;	C^{1}_5+C^{1}_3; 2C^{1}_3+B^{1}_2; A^{2}_5+B^{1}_2;
\(A^{75}_1\)	(1, 0, 1, 1, 0, 1, 1, 0)	(10, 18, 26, 32, 36, 40, 42, 21)	\(V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi} +3V_{0}\)	3	not computed	150	75	A^{2}_4+C^{1}_3;	A^{2}_4+C^{1}_3;
\(A^{72}_1\)	(0, 2, 0, 2, 0, 0, 0, 2)	(10, 20, 26, 32, 34, 36, 38, 20)	\(3V_{10\psi}+V_{8\psi}+7V_{6\psi}+5V_{4\psi}+6V_{2\psi}+2V_{0}\)	2	\(\displaystyle 0\)	144	72	2C^{1}_4; C^{1}_4+C^{1}_3+A^{1}_1; 2C^{1}_3+2A^{1}_1; A^{2}_5+2A^{1}_1; 2C^{1}_3+A^{2}_1; A^{2}_5+A^{2}_1;	2C^{1}_4; C^{1}_4+C^{1}_3+A^{1}_1; 2C^{1}_3+2A^{1}_1; A^{2}_5+2A^{1}_1; 2C^{1}_3+A^{2}_1; A^{2}_5+A^{2}_1;
\(A^{71}_1\)	(0, 2, 0, 2, 0, 0, 1, 0)	(10, 20, 26, 32, 34, 36, 38, 19)	\(3V_{10\psi}+V_{8\psi}+5V_{6\psi}+4V_{5\psi}+3V_{4\psi}+4V_{2\psi}+2V_{\psi}+4V_{0}\)	4	\(\displaystyle A^{1}_1\)	142	71	C^{1}_4+C^{1}_3; 2C^{1}_3+A^{1}_1; A^{2}_5+A^{1}_1;	C^{1}_4+C^{1}_3; 2C^{1}_3+A^{1}_1; A^{2}_5+A^{1}_1;
\(A^{70}_1\)	(0, 2, 0, 2, 0, 1, 0, 0)	(10, 20, 26, 32, 34, 36, 36, 18)	\(3V_{10\psi}+V_{8\psi}+3V_{6\psi}+8V_{5\psi}+V_{4\psi}+3V_{2\psi}+11V_{0}\)	11	\(\displaystyle B^{1}_2\)	140	70	2C^{1}_3; A^{2}_5;	2C^{1}_3; A^{2}_5;
\(A^{56}_1\)	(2, 0, 0, 2, 0, 0, 0, 2)	(10, 16, 22, 28, 30, 32, 34, 18)	\(V_{10\psi}+2V_{8\psi}+7V_{6\psi}+6V_{4\psi}+9V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	112	56	C^{1}_6+B^{1}_2; C^{1}_5+C^{1}_3; C^{1}_5+B^{1}_2+A^{1}_1; C^{1}_4+2B^{1}_2; 2C^{1}_3+B^{1}_2; C^{1}_3+2B^{1}_2+A^{1}_1; C^{1}_4+A^{2}_3; A^{2}_3+C^{1}_3+A^{1}_1;	C^{1}_6+B^{1}_2; C^{1}_5+C^{1}_3; C^{1}_5+B^{1}_2+A^{1}_1; C^{1}_4+2B^{1}_2; 2C^{1}_3+B^{1}_2; 2C^{1}_3+B^{1}_2; C^{1}_3+2B^{1}_2+A^{1}_1; C^{1}_4+A^{2}_3; A^{2}_3+C^{1}_3+A^{1}_1;
\(A^{55}_1\)	(2, 0, 0, 2, 0, 0, 1, 0)	(10, 16, 22, 28, 30, 32, 34, 17)	\(V_{10\psi}+2V_{8\psi}+6V_{6\psi}+2V_{5\psi}+3V_{4\psi}+4V_{3\psi}+6V_{2\psi}+4V_{0}\)	4	\(\displaystyle A^{1}_1\)	110	55	C^{1}_5+B^{1}_2; C^{1}_3+2B^{1}_2; A^{2}_3+C^{1}_3;	C^{1}_5+B^{1}_2; C^{1}_3+2B^{1}_2; A^{2}_3+C^{1}_3;
\(A^{53}_1\)	(2, 0, 1, 0, 1, 0, 1, 0)	(10, 16, 22, 26, 30, 32, 34, 17)	\(V_{10\psi}+V_{8\psi}+2V_{7\psi}+3V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)	3	not computed	106	53	C^{1}_5+A^{2}_2; C^{1}_3+A^{2}_2+B^{1}_2;	C^{1}_5+A^{2}_2; C^{1}_3+A^{2}_2+B^{1}_2;
\(A^{51}_1\)	(0, 1, 1, 0, 1, 0, 1, 0)	(8, 16, 22, 26, 30, 32, 34, 17)	\(3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)	3	not computed	102	51	A^{2}_4+C^{1}_3; A^{2}_4+B^{1}_2+A^{1}_1;	A^{2}_4+C^{1}_3; A^{2}_4+B^{1}_2+A^{1}_1;
\(A^{50}_1\)	(0, 1, 1, 0, 1, 1, 0, 0)	(8, 16, 22, 26, 30, 32, 32, 16)	\(3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+7V_{4\psi}+4V_{3\psi}+2V_{2\psi}+2V_{\psi}+6V_{0}\)	6	not computed	100	50	A^{2}_4+B^{1}_2;	A^{2}_4+B^{1}_2;
\(A^{48}_1\)	(2, 0, 2, 0, 0, 0, 0, 2)	(10, 16, 22, 24, 26, 28, 30, 16)	\(V_{10\psi}+V_{8\psi}+6V_{6\psi}+7V_{4\psi}+12V_{2\psi}+3V_{0}\)	3	not computed	96	48	C^{1}_6+2A^{1}_1; C^{1}_5+3A^{1}_1; C^{1}_4+C^{1}_3+A^{1}_1; C^{1}_4+B^{1}_2+2A^{1}_1; 2C^{1}_3+2A^{1}_1; C^{1}_3+B^{1}_2+3A^{1}_1; C^{1}_6+A^{2}_1; C^{1}_5+A^{2}_1+A^{1}_1; C^{1}_4+B^{1}_2+A^{2}_1; 2C^{1}_3+A^{2}_1; C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1;	C^{1}_6+2A^{1}_1; C^{1}_5+3A^{1}_1; C^{1}_4+C^{1}_3+A^{1}_1; C^{1}_4+B^{1}_2+2A^{1}_1; 2C^{1}_3+2A^{1}_1; 2C^{1}_3+2A^{1}_1; C^{1}_3+B^{1}_2+3A^{1}_1; C^{1}_6+A^{2}_1; C^{1}_5+A^{2}_1+A^{1}_1; C^{1}_4+B^{1}_2+A^{2}_1; 2C^{1}_3+A^{2}_1; 2C^{1}_3+A^{2}_1; C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1;
\(A^{48}_1\)	(0, 2, 0, 0, 0, 2, 0, 0)	(8, 16, 20, 24, 28, 32, 32, 16)	\(3V_{8\psi}+5V_{6\psi}+10V_{4\psi}+6V_{2\psi}+6V_{0}\)	6	not computed	96	48	A^{2}_4+A^{2}_2;	A^{2}_4+A^{2}_2;
\(A^{47}_1\)	(2, 0, 2, 0, 0, 0, 1, 0)	(10, 16, 22, 24, 26, 28, 30, 15)	\(V_{10\psi}+V_{8\psi}+5V_{6\psi}+2V_{5\psi}+5V_{4\psi}+2V_{3\psi}+8V_{2\psi}+4V_{\psi}+4V_{0}\)	4	\(\displaystyle A^{1}_1\)	94	47	C^{1}_6+A^{1}_1; C^{1}_5+2A^{1}_1; C^{1}_4+C^{1}_3; C^{1}_4+B^{1}_2+A^{1}_1; 2C^{1}_3+A^{1}_1; C^{1}_3+B^{1}_2+2A^{1}_1; C^{1}_5+A^{2}_1; C^{1}_3+B^{1}_2+A^{2}_1;	C^{1}_6+A^{1}_1; C^{1}_5+2A^{1}_1; C^{1}_4+C^{1}_3; C^{1}_4+B^{1}_2+A^{1}_1; 2C^{1}_3+A^{1}_1; 2C^{1}_3+A^{1}_1; C^{1}_3+B^{1}_2+2A^{1}_1; C^{1}_5+A^{2}_1; C^{1}_3+B^{1}_2+A^{2}_1;
\(A^{46}_1\)	(2, 0, 2, 0, 0, 1, 0, 0)	(10, 16, 22, 24, 26, 28, 28, 14)	\(V_{10\psi}+V_{8\psi}+4V_{6\psi}+4V_{5\psi}+3V_{4\psi}+4V_{3\psi}+5V_{2\psi}+4V_{\psi}+10V_{0}\)	10	\(\displaystyle B^{1}_2\)	92	46	C^{1}_6; C^{1}_5+A^{1}_1; C^{1}_4+B^{1}_2; 2C^{1}_3; C^{1}_3+B^{1}_2+A^{1}_1;	C^{1}_6; C^{1}_5+A^{1}_1; C^{1}_4+B^{1}_2; 2C^{1}_3; 2C^{1}_3; C^{1}_3+B^{1}_2+A^{1}_1;
\(A^{45}_1\)	(2, 1, 0, 1, 0, 0, 1, 0)	(10, 16, 20, 24, 26, 28, 30, 15)	\(V_{10\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+5V_{4\psi}+6V_{3\psi}+5V_{2\psi}+4V_{\psi}+4V_{0}\)	4	not computed	90	45	C^{1}_4+A^{2}_2+A^{1}_1; C^{1}_3+A^{2}_2+2A^{1}_1; C^{1}_3+A^{2}_2+A^{2}_1;	C^{1}_4+A^{2}_2+A^{1}_1; C^{1}_3+A^{2}_2+2A^{1}_1; C^{1}_3+A^{2}_2+A^{2}_1;
\(A^{45}_1\)	(2, 0, 2, 0, 1, 0, 0, 0)	(10, 16, 22, 24, 26, 26, 26, 13)	\(V_{10\psi}+V_{8\psi}+3V_{6\psi}+6V_{5\psi}+V_{4\psi}+6V_{3\psi}+3V_{2\psi}+21V_{0}\)	21	\(\displaystyle C^{1}_3\)	90	45	C^{1}_5; C^{1}_3+B^{1}_2;	C^{1}_5; C^{1}_3+B^{1}_2;
\(A^{44}_1\)	(2, 1, 0, 1, 0, 1, 0, 0)	(10, 16, 20, 24, 26, 28, 28, 14)	\(V_{10\psi}+2V_{7\psi}+2V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+6V_{0}\)	6	not computed	88	44	C^{1}_4+A^{2}_2; C^{1}_3+A^{2}_2+A^{1}_1;	C^{1}_4+A^{2}_2; C^{1}_3+A^{2}_2+A^{1}_1;
\(A^{43}_1\)	(2, 1, 0, 1, 1, 0, 0, 0)	(10, 16, 20, 24, 26, 26, 26, 13)	\(V_{10\psi}+2V_{7\psi}+V_{6\psi}+6V_{5\psi}+3V_{4\psi}+2V_{3\psi}+10V_{2\psi}+13V_{0}\)	13	not computed	86	43	C^{1}_3+A^{2}_2;	C^{1}_3+A^{2}_2;
\(A^{43}_1\)	(0, 2, 0, 1, 0, 0, 1, 0)	(8, 16, 20, 24, 26, 28, 30, 15)	\(3V_{8\psi}+V_{6\psi}+6V_{5\psi}+3V_{4\psi}+6V_{3\psi}+7V_{2\psi}+6V_{0}\)	6	not computed	86	43	A^{2}_4+3A^{1}_1; A^{2}_4+A^{2}_1+A^{1}_1;	A^{2}_4+3A^{1}_1; A^{2}_4+A^{2}_1+A^{1}_1;
\(A^{42}_1\)	(0, 2, 0, 1, 0, 1, 0, 0)	(8, 16, 20, 24, 26, 28, 28, 14)	\(3V_{8\psi}+V_{6\psi}+4V_{5\psi}+7V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+7V_{0}\)	7	not computed	84	42	A^{2}_4+2A^{1}_1; A^{2}_4+A^{2}_1;	A^{2}_4+2A^{1}_1; A^{2}_4+A^{2}_1;
\(A^{41}_1\)	(0, 2, 0, 1, 1, 0, 0, 0)	(8, 16, 20, 24, 26, 26, 26, 13)	\(3V_{8\psi}+V_{6\psi}+2V_{5\psi}+11V_{4\psi}+2V_{3\psi}+2V_{2\psi}+4V_{\psi}+13V_{0}\)	13	not computed	82	41	A^{2}_4+A^{1}_1;	A^{2}_4+A^{1}_1;
\(A^{40}_1\)	(2, 2, 0, 0, 0, 0, 0, 2)	(10, 16, 18, 20, 22, 24, 26, 14)	\(V_{10\psi}+6V_{6\psi}+5V_{4\psi}+16V_{2\psi}+10V_{0}\)	10	not computed	80	40	C^{1}_4+4A^{1}_1; C^{1}_3+5A^{1}_1; C^{1}_4+A^{2}_1+2A^{1}_1; C^{1}_3+A^{2}_1+3A^{1}_1; C^{1}_4+2A^{2}_1; C^{1}_3+2A^{2}_1+A^{1}_1;	C^{1}_4+4A^{1}_1; C^{1}_3+5A^{1}_1; C^{1}_4+A^{2}_1+2A^{1}_1; C^{1}_3+A^{2}_1+3A^{1}_1; C^{1}_4+2A^{2}_1; C^{1}_3+2A^{2}_1+A^{1}_1;
\(A^{40}_1\)	(0, 0, 0, 2, 0, 0, 0, 2)	(6, 12, 18, 24, 26, 28, 30, 16)	\(10V_{6\psi}+6V_{4\psi}+10V_{2\psi}+6V_{0}\)	6	not computed	80	40	4B^{1}_2; A^{2}_3+2B^{1}_2; 2A^{2}_3;	4B^{1}_2; A^{2}_3+2B^{1}_2; 2A^{2}_3;
\(A^{40}_1\)	(0, 2, 0, 2, 0, 0, 0, 0)	(8, 16, 20, 24, 24, 24, 24, 12)	\(3V_{8\psi}+V_{6\psi}+15V_{4\psi}+V_{2\psi}+24V_{0}\)	24	not computed	80	40	A^{2}_4;	A^{2}_4;
\(A^{39}_1\)	(2, 2, 0, 0, 0, 0, 1, 0)	(10, 16, 18, 20, 22, 24, 26, 13)	\(V_{10\psi}+5V_{6\psi}+2V_{5\psi}+4V_{4\psi}+11V_{2\psi}+8V_{\psi}+9V_{0}\)	9	not computed	78	39	C^{1}_4+3A^{1}_1; C^{1}_3+4A^{1}_1; C^{1}_4+A^{2}_1+A^{1}_1; C^{1}_3+A^{2}_1+2A^{1}_1; C^{1}_3+2A^{2}_1;	C^{1}_4+3A^{1}_1; C^{1}_3+4A^{1}_1; C^{1}_4+A^{2}_1+A^{1}_1; C^{1}_3+A^{2}_1+2A^{1}_1; C^{1}_3+2A^{2}_1;
\(A^{38}_1\)	(2, 2, 0, 0, 0, 1, 0, 0)	(10, 16, 18, 20, 22, 24, 24, 12)	\(V_{10\psi}+4V_{6\psi}+4V_{5\psi}+3V_{4\psi}+7V_{2\psi}+12V_{\psi}+13V_{0}\)	13	not computed	76	38	C^{1}_4+2A^{1}_1; C^{1}_3+3A^{1}_1; C^{1}_4+A^{2}_1; C^{1}_3+A^{2}_1+A^{1}_1;	C^{1}_4+2A^{1}_1; C^{1}_3+3A^{1}_1; C^{1}_4+A^{2}_1; C^{1}_3+A^{2}_1+A^{1}_1;
\(A^{37}_1\)	(2, 2, 0, 0, 1, 0, 0, 0)	(10, 16, 18, 20, 22, 22, 22, 11)	\(V_{10\psi}+3V_{6\psi}+6V_{5\psi}+2V_{4\psi}+4V_{2\psi}+12V_{\psi}+22V_{0}\)	22	\(\displaystyle C^{1}_3\)	74	37	C^{1}_4+A^{1}_1; C^{1}_3+2A^{1}_1; C^{1}_3+A^{2}_1;	C^{1}_4+A^{1}_1; C^{1}_3+2A^{1}_1; C^{1}_3+A^{2}_1;
\(A^{36}_1\)	(2, 2, 0, 1, 0, 0, 0, 0)	(10, 16, 18, 20, 20, 20, 20, 10)	\(V_{10\psi}+2V_{6\psi}+8V_{5\psi}+V_{4\psi}+2V_{2\psi}+8V_{\psi}+36V_{0}\)	36	\(\displaystyle C^{1}_4\)	72	36	C^{1}_4; C^{1}_3+A^{1}_1;	C^{1}_4; C^{1}_3+A^{1}_1;
\(A^{35}_1\)	(2, 2, 1, 0, 0, 0, 0, 0)	(10, 16, 18, 18, 18, 18, 18, 9)	\(V_{10\psi}+V_{6\psi}+10V_{5\psi}+V_{2\psi}+55V_{0}\)	55	\(\displaystyle C^{1}_5\)	70	35	C^{1}_3;	C^{1}_3;
\(A^{32}_1\)	(0, 0, 2, 0, 0, 0, 0, 2)	(6, 12, 18, 20, 22, 24, 26, 14)	\(6V_{6\psi}+9V_{4\psi}+15V_{2\psi}+4V_{0}\)	4	not computed	64	32	2C^{1}_3+B^{1}_2; C^{1}_3+2B^{1}_2+A^{1}_1; 3B^{1}_2+2A^{1}_1; A^{2}_3+C^{1}_3+A^{1}_1; A^{2}_3+B^{1}_2+2A^{1}_1; 3B^{1}_2+A^{2}_1; A^{2}_3+B^{1}_2+A^{2}_1;	2C^{1}_3+B^{1}_2; C^{1}_3+2B^{1}_2+A^{1}_1; 3B^{1}_2+2A^{1}_1; A^{2}_3+C^{1}_3+A^{1}_1; A^{2}_3+B^{1}_2+2A^{1}_1; 3B^{1}_2+A^{2}_1; A^{2}_3+B^{1}_2+A^{2}_1;
\(A^{31}_1\)	(0, 0, 2, 0, 0, 0, 1, 0)	(6, 12, 18, 20, 22, 24, 26, 13)	\(6V_{6\psi}+6V_{4\psi}+6V_{3\psi}+10V_{2\psi}+2V_{\psi}+6V_{0}\)	6	not computed	62	31	C^{1}_3+2B^{1}_2; 3B^{1}_2+A^{1}_1; A^{2}_3+C^{1}_3; A^{2}_3+B^{1}_2+A^{1}_1;	C^{1}_3+2B^{1}_2; 3B^{1}_2+A^{1}_1; A^{2}_3+C^{1}_3; A^{2}_3+B^{1}_2+A^{1}_1;
\(A^{30}_1\)	(0, 0, 2, 0, 0, 1, 0, 0)	(6, 12, 18, 20, 22, 24, 24, 12)	\(6V_{6\psi}+3V_{4\psi}+12V_{3\psi}+6V_{2\psi}+13V_{0}\)	13	not computed	60	30	3B^{1}_2; A^{2}_3+B^{1}_2;	3B^{1}_2; A^{2}_3+B^{1}_2;
\(A^{29}_1\)	(0, 1, 0, 1, 0, 0, 1, 0)	(6, 12, 16, 20, 22, 24, 26, 13)	\(3V_{6\psi}+4V_{5\psi}+6V_{4\psi}+6V_{3\psi}+7V_{2\psi}+6V_{\psi}+4V_{0}\)	4	not computed	58	29	C^{1}_3+A^{2}_2+B^{1}_2; A^{2}_2+2B^{1}_2+A^{1}_1; A^{2}_3+A^{2}_2+A^{1}_1;	C^{1}_3+A^{2}_2+B^{1}_2; A^{2}_2+2B^{1}_2+A^{1}_1; A^{2}_3+A^{2}_2+A^{1}_1;
\(A^{28}_1\)	(0, 1, 0, 1, 0, 1, 0, 0)	(6, 12, 16, 20, 22, 24, 24, 12)	\(3V_{6\psi}+4V_{5\psi}+4V_{4\psi}+8V_{3\psi}+8V_{2\psi}+4V_{\psi}+7V_{0}\)	7	not computed	56	28	A^{2}_2+2B^{1}_2; A^{2}_3+A^{2}_2;	A^{2}_2+2B^{1}_2; A^{2}_3+A^{2}_2;
\(A^{26}_1\)	(1, 0, 0, 0, 1, 1, 0, 0)	(6, 10, 14, 18, 22, 24, 24, 12)	\(V_{6\psi}+4V_{5\psi}+10V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+10V_{0}\)	10	not computed	52	26	2A^{2}_2+B^{1}_2;	2A^{2}_2+B^{1}_2;
\(A^{24}_1\)	(0, 2, 0, 0, 0, 0, 0, 2)	(6, 12, 14, 16, 18, 20, 22, 12)	\(3V_{6\psi}+9V_{4\psi}+21V_{2\psi}+7V_{0}\)	7	not computed	48	24	2C^{1}_3+2A^{1}_1; C^{1}_3+B^{1}_2+3A^{1}_1; 2B^{1}_2+4A^{1}_1; A^{2}_3+4A^{1}_1; 2C^{1}_3+A^{2}_1; C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1; 2B^{1}_2+A^{2}_1+2A^{1}_1; A^{2}_3+A^{2}_1+2A^{1}_1; 2B^{1}_2+2A^{2}_1; A^{2}_3+2A^{2}_1;	2C^{1}_3+2A^{1}_1; C^{1}_3+B^{1}_2+3A^{1}_1; 2B^{1}_2+4A^{1}_1; A^{2}_3+4A^{1}_1; 2C^{1}_3+A^{2}_1; C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1; 2B^{1}_2+A^{2}_1+2A^{1}_1; A^{2}_3+A^{2}_1+2A^{1}_1; 2B^{1}_2+2A^{2}_1; A^{2}_3+2A^{2}_1;
\(A^{23}_1\)	(0, 2, 0, 0, 0, 0, 1, 0)	(6, 12, 14, 16, 18, 20, 22, 11)	\(3V_{6\psi}+7V_{4\psi}+4V_{3\psi}+15V_{2\psi}+6V_{\psi}+7V_{0}\)	7	not computed	46	23	2C^{1}_3+A^{1}_1; C^{1}_3+B^{1}_2+2A^{1}_1; 2B^{1}_2+3A^{1}_1; A^{2}_3+3A^{1}_1; C^{1}_3+B^{1}_2+A^{2}_1; 2B^{1}_2+A^{2}_1+A^{1}_1; A^{2}_3+A^{2}_1+A^{1}_1;	2C^{1}_3+A^{1}_1; C^{1}_3+B^{1}_2+2A^{1}_1; 2B^{1}_2+3A^{1}_1; A^{2}_3+3A^{1}_1; C^{1}_3+B^{1}_2+A^{2}_1; 2B^{1}_2+A^{2}_1+A^{1}_1; A^{2}_3+A^{2}_1+A^{1}_1;
\(A^{22}_1\)	(0, 2, 0, 0, 0, 1, 0, 0)	(6, 12, 14, 16, 18, 20, 20, 10)	\(3V_{6\psi}+5V_{4\psi}+8V_{3\psi}+10V_{2\psi}+8V_{\psi}+12V_{0}\)	12	\(\displaystyle B^{1}_2\)	44	22	2C^{1}_3; C^{1}_3+B^{1}_2+A^{1}_1; 2B^{1}_2+2A^{1}_1; A^{2}_3+2A^{1}_1; 2B^{1}_2+A^{2}_1; A^{2}_3+A^{2}_1;	2C^{1}_3; C^{1}_3+B^{1}_2+A^{1}_1; 2B^{1}_2+2A^{1}_1; A^{2}_3+2A^{1}_1; 2B^{1}_2+A^{2}_1; A^{2}_3+A^{2}_1;
\(A^{21}_1\)	(1, 0, 1, 0, 0, 0, 1, 0)	(6, 10, 14, 16, 18, 20, 22, 11)	\(V_{6\psi}+2V_{5\psi}+6V_{4\psi}+8V_{3\psi}+11V_{2\psi}+8V_{\psi}+6V_{0}\)	6	not computed	42	21	C^{1}_3+A^{2}_2+2A^{1}_1; A^{2}_2+B^{1}_2+3A^{1}_1; C^{1}_3+A^{2}_2+A^{2}_1; A^{2}_2+B^{1}_2+A^{2}_1+A^{1}_1;	C^{1}_3+A^{2}_2+2A^{1}_1; A^{2}_2+B^{1}_2+3A^{1}_1; C^{1}_3+A^{2}_2+A^{2}_1; A^{2}_2+B^{1}_2+A^{2}_1+A^{1}_1;
\(A^{21}_1\)	(0, 2, 0, 0, 1, 0, 0, 0)	(6, 12, 14, 16, 18, 18, 18, 9)	\(3V_{6\psi}+3V_{4\psi}+12V_{3\psi}+6V_{2\psi}+6V_{\psi}+22V_{0}\)	22	\(\displaystyle C^{1}_3\)	42	21	C^{1}_3+B^{1}_2; 2B^{1}_2+A^{1}_1; A^{2}_3+A^{1}_1;	C^{1}_3+B^{1}_2; 2B^{1}_2+A^{1}_1; A^{2}_3+A^{1}_1;
\(A^{20}_1\)	(1, 0, 1, 0, 0, 1, 0, 0)	(6, 10, 14, 16, 18, 20, 20, 10)	\(V_{6\psi}+2V_{5\psi}+5V_{4\psi}+8V_{3\psi}+11V_{2\psi}+10V_{\psi}+7V_{0}\)	7	not computed	40	20	C^{1}_3+A^{2}_2+A^{1}_1; A^{2}_2+B^{1}_2+2A^{1}_1; A^{2}_2+B^{1}_2+A^{2}_1;	C^{1}_3+A^{2}_2+A^{1}_1; A^{2}_2+B^{1}_2+2A^{1}_1; A^{2}_2+B^{1}_2+A^{2}_1;
\(A^{20}_1\)	(0, 2, 0, 1, 0, 0, 0, 0)	(6, 12, 14, 16, 16, 16, 16, 8)	\(3V_{6\psi}+V_{4\psi}+16V_{3\psi}+3V_{2\psi}+37V_{0}\)	37	\(\displaystyle C^{1}_4\)	40	20	2B^{1}_2; A^{2}_3;	2B^{1}_2; A^{2}_3;
\(A^{19}_1\)	(1, 0, 1, 0, 1, 0, 0, 0)	(6, 10, 14, 16, 18, 18, 18, 9)	\(V_{6\psi}+2V_{5\psi}+4V_{4\psi}+8V_{3\psi}+12V_{2\psi}+8V_{\psi}+13V_{0}\)	13	not computed	38	19	C^{1}_3+A^{2}_2; A^{2}_2+B^{1}_2+A^{1}_1;	C^{1}_3+A^{2}_2; A^{2}_2+B^{1}_2+A^{1}_1;
\(A^{18}_1\)	(1, 0, 1, 1, 0, 0, 0, 0)	(6, 10, 14, 16, 16, 16, 16, 8)	\(V_{6\psi}+2V_{5\psi}+3V_{4\psi}+8V_{3\psi}+14V_{2\psi}+2V_{\psi}+24V_{0}\)	24	not computed	36	18	A^{2}_2+B^{1}_2;	A^{2}_2+B^{1}_2;
\(A^{18}_1\)	(0, 0, 0, 1, 0, 1, 0, 0)	(4, 8, 12, 16, 18, 20, 20, 10)	\(10V_{4\psi}+8V_{3\psi}+9V_{2\psi}+8V_{\psi}+11V_{0}\)	11	not computed	36	18	2A^{2}_2+2A^{1}_1; 2A^{2}_2+A^{2}_1;	2A^{2}_2+2A^{1}_1; 2A^{2}_2+A^{2}_1;
\(A^{17}_1\)	(0, 0, 0, 1, 1, 0, 0, 0)	(4, 8, 12, 16, 18, 18, 18, 9)	\(10V_{4\psi}+4V_{3\psi}+15V_{2\psi}+6V_{\psi}+13V_{0}\)	13	not computed	34	17	2A^{2}_2+A^{1}_1;	2A^{2}_2+A^{1}_1;
\(A^{16}_1\)	(2, 0, 0, 0, 0, 0, 0, 2)	(6, 8, 10, 12, 14, 16, 18, 10)	\(V_{6\psi}+6V_{4\psi}+28V_{2\psi}+15V_{0}\)	15	not computed	32	16	C^{1}_3+5A^{1}_1; B^{1}_2+6A^{1}_1; C^{1}_3+A^{2}_1+3A^{1}_1; B^{1}_2+A^{2}_1+4A^{1}_1; C^{1}_3+2A^{2}_1+A^{1}_1; B^{1}_2+2A^{2}_1+2A^{1}_1; B^{1}_2+3A^{2}_1;	C^{1}_3+5A^{1}_1; B^{1}_2+6A^{1}_1; C^{1}_3+A^{2}_1+3A^{1}_1; B^{1}_2+A^{2}_1+4A^{1}_1; C^{1}_3+2A^{2}_1+A^{1}_1; B^{1}_2+2A^{2}_1+2A^{1}_1; B^{1}_2+3A^{2}_1;
\(A^{16}_1\)	(0, 0, 0, 2, 0, 0, 0, 0)	(4, 8, 12, 16, 16, 16, 16, 8)	\(10V_{4\psi}+22V_{2\psi}+20V_{0}\)	20	not computed	32	16	2A^{2}_2;	2A^{2}_2;
\(A^{15}_1\)	(2, 0, 0, 0, 0, 0, 1, 0)	(6, 8, 10, 12, 14, 16, 18, 9)	\(V_{6\psi}+5V_{4\psi}+2V_{3\psi}+21V_{2\psi}+10V_{\psi}+13V_{0}\)	13	not computed	30	15	C^{1}_3+4A^{1}_1; B^{1}_2+5A^{1}_1; C^{1}_3+A^{2}_1+2A^{1}_1; B^{1}_2+A^{2}_1+3A^{1}_1; C^{1}_3+2A^{2}_1; B^{1}_2+2A^{2}_1+A^{1}_1;	C^{1}_3+4A^{1}_1; B^{1}_2+5A^{1}_1; C^{1}_3+A^{2}_1+2A^{1}_1; B^{1}_2+A^{2}_1+3A^{1}_1; C^{1}_3+2A^{2}_1; B^{1}_2+2A^{2}_1+A^{1}_1;
\(A^{14}_1\)	(2, 0, 0, 0, 0, 1, 0, 0)	(6, 8, 10, 12, 14, 16, 16, 8)	\(V_{6\psi}+4V_{4\psi}+4V_{3\psi}+15V_{2\psi}+16V_{\psi}+16V_{0}\)	16	not computed	28	14	C^{1}_3+3A^{1}_1; B^{1}_2+4A^{1}_1; C^{1}_3+A^{2}_1+A^{1}_1; B^{1}_2+A^{2}_1+2A^{1}_1; B^{1}_2+2A^{2}_1;	C^{1}_3+3A^{1}_1; B^{1}_2+4A^{1}_1; C^{1}_3+A^{2}_1+A^{1}_1; B^{1}_2+A^{2}_1+2A^{1}_1; B^{1}_2+2A^{2}_1;
\(A^{13}_1\)	(2, 0, 0, 0, 1, 0, 0, 0)	(6, 8, 10, 12, 14, 14, 14, 7)	\(V_{6\psi}+3V_{4\psi}+6V_{3\psi}+10V_{2\psi}+18V_{\psi}+24V_{0}\)	24	not computed	26	13	C^{1}_3+2A^{1}_1; B^{1}_2+3A^{1}_1; C^{1}_3+A^{2}_1; B^{1}_2+A^{2}_1+A^{1}_1;	C^{1}_3+2A^{1}_1; B^{1}_2+3A^{1}_1; C^{1}_3+A^{2}_1; B^{1}_2+A^{2}_1+A^{1}_1;
\(A^{13}_1\)	(0, 1, 0, 0, 0, 0, 1, 0)	(4, 8, 10, 12, 14, 16, 18, 9)	\(3V_{4\psi}+10V_{3\psi}+16V_{2\psi}+10V_{\psi}+13V_{0}\)	13	not computed	26	13	A^{2}_2+5A^{1}_1; A^{2}_2+A^{2}_1+3A^{1}_1; A^{2}_2+2A^{2}_1+A^{1}_1;	A^{2}_2+5A^{1}_1; A^{2}_2+A^{2}_1+3A^{1}_1; A^{2}_2+2A^{2}_1+A^{1}_1;
\(A^{12}_1\)	(2, 0, 0, 1, 0, 0, 0, 0)	(6, 8, 10, 12, 12, 12, 12, 6)	\(V_{6\psi}+2V_{4\psi}+8V_{3\psi}+6V_{2\psi}+16V_{\psi}+37V_{0}\)	37	\(\displaystyle C^{1}_4\)	24	12	C^{1}_3+A^{1}_1; B^{1}_2+2A^{1}_1; B^{1}_2+A^{2}_1;	C^{1}_3+A^{1}_1; B^{1}_2+2A^{1}_1; B^{1}_2+A^{2}_1;
\(A^{12}_1\)	(0, 1, 0, 0, 0, 1, 0, 0)	(4, 8, 10, 12, 14, 16, 16, 8)	\(3V_{4\psi}+8V_{3\psi}+15V_{2\psi}+16V_{\psi}+12V_{0}\)	12	not computed	24	12	A^{2}_2+4A^{1}_1; A^{2}_2+A^{2}_1+2A^{1}_1; A^{2}_2+2A^{2}_1;	A^{2}_2+4A^{1}_1; A^{2}_2+A^{2}_1+2A^{1}_1; A^{2}_2+2A^{2}_1;
\(A^{11}_1\)	(2, 0, 1, 0, 0, 0, 0, 0)	(6, 8, 10, 10, 10, 10, 10, 5)	\(V_{6\psi}+V_{4\psi}+10V_{3\psi}+3V_{2\psi}+10V_{\psi}+55V_{0}\)	55	\(\displaystyle C^{1}_5\)	22	11	C^{1}_3; B^{1}_2+A^{1}_1;	C^{1}_3; B^{1}_2+A^{1}_1;
\(A^{11}_1\)	(0, 1, 0, 0, 1, 0, 0, 0)	(4, 8, 10, 12, 14, 14, 14, 7)	\(3V_{4\psi}+6V_{3\psi}+15V_{2\psi}+18V_{\psi}+16V_{0}\)	16	not computed	22	11	A^{2}_2+3A^{1}_1; A^{2}_2+A^{2}_1+A^{1}_1;	A^{2}_2+3A^{1}_1; A^{2}_2+A^{2}_1+A^{1}_1;
\(A^{10}_1\)	(2, 1, 0, 0, 0, 0, 0, 0)	(6, 8, 8, 8, 8, 8, 8, 4)	\(V_{6\psi}+12V_{3\psi}+V_{2\psi}+78V_{0}\)	78	\(\displaystyle C^{1}_6\)	20	10	B^{1}_2;	B^{1}_2;
\(A^{10}_1\)	(0, 1, 0, 1, 0, 0, 0, 0)	(4, 8, 10, 12, 12, 12, 12, 6)	\(3V_{4\psi}+4V_{3\psi}+16V_{2\psi}+16V_{\psi}+25V_{0}\)	25	not computed	20	10	A^{2}_2+2A^{1}_1; A^{2}_2+A^{2}_1;	A^{2}_2+2A^{1}_1; A^{2}_2+A^{2}_1;
\(A^{9}_1\)	(0, 1, 1, 0, 0, 0, 0, 0)	(4, 8, 10, 10, 10, 10, 10, 5)	\(3V_{4\psi}+2V_{3\psi}+18V_{2\psi}+10V_{\psi}+39V_{0}\)	39	not computed	18	9	A^{2}_2+A^{1}_1;	A^{2}_2+A^{1}_1;
\(A^{8}_1\)	(0, 0, 0, 0, 0, 0, 0, 2)	(2, 4, 6, 8, 10, 12, 14, 8)	\(36V_{2\psi}+28V_{0}\)	28	not computed	16	8	8A^{1}_1; A^{2}_1+6A^{1}_1; 2A^{2}_1+4A^{1}_1; 3A^{2}_1+2A^{1}_1; 4A^{2}_1;	8A^{1}_1; A^{2}_1+6A^{1}_1; 2A^{2}_1+4A^{1}_1; 3A^{2}_1+2A^{1}_1; 4A^{2}_1;
\(A^{8}_1\)	(0, 2, 0, 0, 0, 0, 0, 0)	(4, 8, 8, 8, 8, 8, 8, 4)	\(3V_{4\psi}+21V_{2\psi}+58V_{0}\)	58	not computed	16	8	A^{2}_2;	A^{2}_2;
\(A^{7}_1\)	(0, 0, 0, 0, 0, 0, 1, 0)	(2, 4, 6, 8, 10, 12, 14, 7)	\(28V_{2\psi}+14V_{\psi}+24V_{0}\)	24	not computed	14	7	7A^{1}_1; A^{2}_1+5A^{1}_1; 2A^{2}_1+3A^{1}_1; 3A^{2}_1+A^{1}_1;	7A^{1}_1; A^{2}_1+5A^{1}_1; 2A^{2}_1+3A^{1}_1; 3A^{2}_1+A^{1}_1;
\(A^{6}_1\)	(0, 0, 0, 0, 0, 1, 0, 0)	(2, 4, 6, 8, 10, 12, 12, 6)	\(21V_{2\psi}+24V_{\psi}+25V_{0}\)	25	not computed	12	6	6A^{1}_1; A^{2}_1+4A^{1}_1; 2A^{2}_1+2A^{1}_1; 3A^{2}_1;	6A^{1}_1; A^{2}_1+4A^{1}_1; 2A^{2}_1+2A^{1}_1; 3A^{2}_1;
\(A^{5}_1\)	(0, 0, 0, 0, 1, 0, 0, 0)	(2, 4, 6, 8, 10, 10, 10, 5)	\(15V_{2\psi}+30V_{\psi}+31V_{0}\)	31	not computed	10	5	5A^{1}_1; A^{2}_1+3A^{1}_1; 2A^{2}_1+A^{1}_1;	5A^{1}_1; A^{2}_1+3A^{1}_1; 2A^{2}_1+A^{1}_1;
\(A^{4}_1\)	(0, 0, 0, 1, 0, 0, 0, 0)	(2, 4, 6, 8, 8, 8, 8, 4)	\(10V_{2\psi}+32V_{\psi}+42V_{0}\)	42	not computed	8	4	4A^{1}_1; A^{2}_1+2A^{1}_1; 2A^{2}_1;	4A^{1}_1; A^{2}_1+2A^{1}_1; 2A^{2}_1;
\(A^{3}_1\)	(0, 0, 1, 0, 0, 0, 0, 0)	(2, 4, 6, 6, 6, 6, 6, 3)	\(6V_{2\psi}+30V_{\psi}+58V_{0}\)	58	not computed	6	3	3A^{1}_1; A^{2}_1+A^{1}_1;	3A^{1}_1; A^{2}_1+A^{1}_1;
\(A^{2}_1\)	(0, 1, 0, 0, 0, 0, 0, 0)	(2, 4, 4, 4, 4, 4, 4, 2)	\(3V_{2\psi}+24V_{\psi}+79V_{0}\)	79	\(\displaystyle C^{1}_6\)	4	2	2A^{1}_1; A^{2}_1;	2A^{1}_1; A^{2}_1;
\(A^{1}_1\)	(1, 0, 0, 0, 0, 0, 0, 0)	(2, 2, 2, 2, 2, 2, 2, 1)	\(V_{2\psi}+14V_{\psi}+105V_{0}\)	105	\(\displaystyle C^{1}_7\)	2	1	A^{1}_1;	A^{1}_1;

Length longest root ambient algebra squared/4= 1/2

Given a root subsystem P, and a root subsubsystem P_0, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant e(P, P_0) (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an sl(2) is an S-subalgebra in the root subalgebra generated by P, such that it has characteristic 2 for all simple roots of P lying in P_0, then e(P, P_0)= 0. It turns out by direct computation that, in the current case of C^{1}_8, e(P,P_0)= 0 implies that an S-sl(2) subalgebra of the root subalgebra generated by P with characteristic with 2's in the simple roots of P_0 always exists. Note that Theorem 10.3 is stated in one direction only.

h-characteristic: (2, 2, 2, 2, 2, 2, 2, 2)
Length of the weight dual to h: 1360
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(V_{30\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 64h_{8}+126h_{7}+120h_{6}+110h_{5}+96h_{4}+78h_{3}+56h_{2}+30h_{1}\)
\( e = 32/25g_{8}+63/37g_{7}+30/13g_{6}+55/17g_{5}+24/5g_{4}+39/5g_{3}+14g_{2}+15g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{9} -30~\\2x_{2} x_{10} -56~\\2x_{3} x_{11} -78~\\2x_{4} x_{12} -96~\\2x_{5} x_{13} -110~\\2x_{6} x_{14} -120~\\2x_{7} x_{15} -126~\\x_{8} x_{16} -64~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 2, 2, 0, 2)
Length of the weight dual to h: 912
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_8 Containing regular semisimple subalgebra number 2: C^{1}_7+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{26\psi}+V_{22\psi}+V_{18\psi}+2V_{14\psi}+V_{12\psi}+V_{10\psi}+V_{6\psi}+2V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 50h_{8}+98h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\)
\( e = 9744298/31455501g_{22}+7909915/31455501g_{15}+21528/26411g_{14}+2243257/31455501g_{8}+14160/26411g_{6}+45/17g_{5}+4g_{4}+33/5g_{3}+12g_{2}+13g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{20} +x_{7} x_{19} -x_{6} x_{18} ~\\x_{10} x_{19} +x_{9} x_{17} -x_{8} x_{16} ~\\2x_{1} x_{11} -26~\\2x_{2} x_{12} -48~\\2x_{3} x_{13} -66~\\2x_{4} x_{14} -80~\\2x_{5} x_{15} -90~\\2x_{8} x_{18} +2x_{6} x_{16} -96~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} -98~\\x_{10} x_{20} +2x_{9} x_{19} +x_{7} x_{17} -50~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 2, 2, 1, 0)
Length of the weight dual to h: 910
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+2V_{13\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 49h_{8}+98h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\)
\( e = 49/37g_{22}+24/13g_{6}+45/17g_{5}+4g_{4}+33/5g_{3}+12g_{2}+13g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{8} -26~\\2x_{2} x_{9} -48~\\2x_{3} x_{10} -66~\\2x_{4} x_{11} -80~\\2x_{5} x_{12} -90~\\2x_{6} x_{13} -96~\\2x_{7} x_{14} -98~\\x_{7} x_{14} -49~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 0, 2, 0, 2)
Length of the weight dual to h: 592
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_8 Containing regular semisimple subalgebra number 2: C^{1}_6+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+V_{18\psi}+2V_{14\psi}+V_{12\psi}+2V_{10\psi}+V_{8\psi}+2V_{6\psi}+2V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 40h_{8}+78h_{7}+76h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\)
\( e = 688499677/3364014692g_{22}+2970113655/11614942619g_{20}+691053783/3364014692g_{15}+2900635205/11614942619g_{14}+3422574634/11614942619g_{13}+26137712/22729829g_{12}+27268653/3364014692g_{8}+231453227/11614942619g_{6}+12593984/22729829g_{4}+27/5g_{3}+10g_{2}+11g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{23} +x_{5} x_{20} -x_{4} x_{19} ~\\x_{10} x_{24} -x_{8} x_{23} +x_{6} x_{22} -x_{5} x_{21} ~\\x_{11} x_{21} +x_{8} x_{17} -x_{7} x_{16} ~\\x_{12} x_{22} -x_{11} x_{20} +x_{10} x_{18} -x_{9} x_{17} ~\\2x_{1} x_{13} -22~\\2x_{2} x_{14} -40~\\2x_{3} x_{15} -54~\\2x_{7} x_{19} +2x_{4} x_{16} -64~\\2x_{9} x_{21} +2x_{5} x_{17} +2x_{4} x_{16} -70~\\2x_{11} x_{23} +2x_{9} x_{21} +2x_{8} x_{20} +2x_{5} x_{17} -76~\\2x_{10} x_{22} +2x_{8} x_{20} +2x_{6} x_{18} +2x_{5} x_{17} -78~\\x_{12} x_{24} +2x_{10} x_{22} +x_{6} x_{18} -40~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 2, 0, 0, 2)
Length of the weight dual to h: 576
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_6+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+V_{18\psi}+V_{14\psi}+2V_{12\psi}+3V_{10\psi}+V_{6\psi}+4V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 38h_{8}+74h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\)
\( e = 1/37g_{34}+83401632/208460707g_{22}+81270/99599g_{20}+2892870/16035439g_{15}+84175/199198g_{13}+1230119/32070878g_{8}+16/5g_{4}+27/5g_{3}+10g_{2}+11g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{20} +x_{6} x_{19} -x_{5} x_{18} ~\\x_{10} x_{19} +x_{9} x_{16} -x_{8} x_{15} ~\\2x_{1} x_{11} -22~\\2x_{2} x_{12} -40~\\2x_{3} x_{13} -54~\\2x_{4} x_{14} -64~\\2x_{8} x_{18} +2x_{5} x_{15} -70~\\2x_{8} x_{18} +2x_{7} x_{17} +2x_{5} x_{15} -72~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} -74~\\x_{10} x_{20} +2x_{9} x_{19} +x_{7} x_{17} +x_{6} x_{16} -38~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 2, 0, 1, 0)
Length of the weight dual to h: 574
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_7 Containing regular semisimple subalgebra number 2: C^{1}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{11\psi}+2V_{10\psi}+V_{6\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 37h_{8}+74h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\)
\( e = 693067/2111022g_{34}+263290/1055511g_{28}+285772/5277555g_{22}+34475/39093g_{13}+21140/39093g_{5}+16/5g_{4}+27/5g_{3}+10g_{2}+11g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{6} x_{17} -x_{5} x_{16} ~\\x_{9} x_{17} +x_{8} x_{15} -x_{7} x_{14} ~\\2x_{1} x_{10} -22~\\2x_{2} x_{11} -40~\\2x_{3} x_{12} -54~\\2x_{4} x_{13} -64~\\2x_{7} x_{16} +2x_{5} x_{14} -70~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} -72~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} -74~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} -37~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 2, 1, 0, 0)
Length of the weight dual to h: 572
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+V_{18\psi}+V_{14\psi}+4V_{11\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 36h_{8}+72h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\)
\( e = 18/13g_{34}+35/17g_{5}+16/5g_{4}+27/5g_{3}+10g_{2}+11g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -22~\\2x_{2} x_{8} -40~\\2x_{3} x_{9} -54~\\2x_{4} x_{10} -64~\\2x_{5} x_{11} -70~\\2x_{6} x_{12} -72~\\2x_{6} x_{12} -72~\\x_{6} x_{12} -36~\\\end{array}\)

h-characteristic: (2, 2, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 400
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_8 Containing regular semisimple subalgebra number 2: C^{1}_5+C^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+2V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+2V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 34h_{8}+66h_{7}+64h_{6}+58h_{5}+52h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 12905306826298/222409966511561g_{22}+446655252447/8852592019541g_{20}+1357491076927/11316715571372g_{18}+38577027558859/222409966511561g_{15}+9527140063466/44262960097705g_{14}+2018225754417/8852592019541g_{13}+2814256268045/11316715571372g_{12}+7645075806057/22633431142744g_{11}+160228534124/91263835253g_{10}-2647123067903/222409966511561g_{8}-72627176458/8852592019541g_{6}-127987337673/22633431142744g_{4}+43837088300/91263835253g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{26} +x_{3} x_{21} -x_{2} x_{20} ~\\x_{10} x_{27} -x_{7} x_{26} +x_{4} x_{23} -x_{3} x_{22} ~\\x_{11} x_{28} -x_{9} x_{27} +x_{5} x_{25} -x_{4} x_{24} ~\\x_{12} x_{22} +x_{7} x_{17} -x_{6} x_{16} ~\\x_{13} x_{24} -x_{12} x_{21} +x_{9} x_{18} -x_{8} x_{17} ~\\x_{14} x_{25} -x_{13} x_{23} +x_{11} x_{19} -x_{10} x_{18} ~\\2x_{1} x_{15} -18~\\2x_{6} x_{20} +2x_{2} x_{16} -32~\\2x_{8} x_{22} +2x_{3} x_{17} +2x_{2} x_{16} -42~\\2x_{12} x_{26} +2x_{8} x_{22} +2x_{7} x_{21} +2x_{3} x_{17} -52~\\2x_{10} x_{24} +2x_{7} x_{21} +2x_{4} x_{18} +2x_{3} x_{17} -58~\\2x_{13} x_{27} +2x_{10} x_{24} +2x_{9} x_{23} +2x_{4} x_{18} -64~\\2x_{11} x_{25} +2x_{9} x_{23} +2x_{5} x_{19} +2x_{4} x_{18} -66~\\x_{14} x_{28} +2x_{11} x_{25} +x_{5} x_{19} -34~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 2, 0, 0, 2)
Length of the weight dual to h: 352
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: C^{1}_8 Containing regular semisimple subalgebra number 2: C^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_6+B^{1}_2 Containing regular semisimple subalgebra number 4: C^{1}_5+C^{1}_3 Containing regular semisimple subalgebra number 5: C^{1}_5+B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{8\psi}+3V_{6\psi}+V_{4\psi}+4V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 30h_{8}+58h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 655091337656062236323364/4132167318366859107188489g_{34}-250143806248769258400196/4132167318366859107188489g_{28}-1561732546028622696/45272614227283633793g_{26}+576410992093490222735650/4132167318366859107188489g_{22}+409124744469626321246186/4132167318366859107188489g_{21}+8861711639174803704/45272614227283633793g_{20}+4174836474562761174/45272614227283633793g_{19}-105478545510948212060769/4132167318366859107188489g_{15}+3120671914332595698/45272614227283633793g_{13}+6273334835818585437/45272614227283633793g_{12}+1218985887/1062052561g_{11}+15485296683015086997289/4132167318366859107188489g_{8}-1106759765727126075/45272614227283633793g_{5}+623391321/1062052561g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{11} x_{30} +x_{8} x_{26} +x_{4} x_{23} -x_{3} x_{22} ~\\x_{15} x_{32} +x_{13} x_{31} +x_{9} x_{28} -x_{7} x_{26} -x_{4} x_{24} ~\\x_{12} x_{32} +x_{9} x_{31} -x_{7} x_{30} +x_{5} x_{28} -x_{4} x_{27} ~\\x_{12} x_{31} -x_{10} x_{30} +x_{9} x_{29} -x_{8} x_{27} +x_{5} x_{25} ~\\x_{14} x_{27} +x_{10} x_{24} +x_{7} x_{20} -x_{6} x_{19} ~\\x_{16} x_{31} +x_{15} x_{29} +x_{12} x_{25} -x_{10} x_{23} -x_{8} x_{20} ~\\x_{15} x_{28} -x_{14} x_{26} +x_{13} x_{25} -x_{11} x_{24} +x_{9} x_{21} ~\\x_{16} x_{28} +x_{15} x_{25} -x_{14} x_{23} +x_{12} x_{21} -x_{11} x_{20} ~\\2x_{1} x_{17} -18~\\2x_{2} x_{18} -32~\\2x_{6} x_{22} +2x_{3} x_{19} -42~\\2x_{11} x_{27} +2x_{8} x_{24} +2x_{4} x_{20} +2x_{3} x_{19} -48~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{4} x_{20} -54~\\2x_{12} x_{28} +2x_{10} x_{26} +2x_{9} x_{25} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} -56~\\2x_{15} x_{31} +2x_{13} x_{29} +2x_{12} x_{28} +4x_{9} x_{25} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} -58~\\x_{16} x_{32} +2x_{15} x_{31} +x_{13} x_{29} +2x_{12} x_{28} +2x_{9} x_{25} +x_{5} x_{21} -30~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 2, 0, 1, 0)
Length of the weight dual to h: 350
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_7 Containing regular semisimple subalgebra number 2: C^{1}_5+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}+3V_{6\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 29h_{8}+58h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 858784739/4229333823g_{34}+839224831/4229333823g_{28}-10375126/4229333823g_{22}+14246403/53443919g_{19}+168706932/694770947g_{13}+157949451/534439190g_{12}+3402357/2594365g_{11}+4852590/694770947g_{5}+3746988/6745349g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{21} +x_{4} x_{18} -x_{3} x_{17} ~\\x_{9} x_{22} -x_{7} x_{21} +x_{5} x_{20} -x_{4} x_{19} ~\\x_{10} x_{19} +x_{7} x_{15} -x_{6} x_{14} ~\\x_{11} x_{20} -x_{10} x_{18} +x_{9} x_{16} -x_{8} x_{15} ~\\2x_{1} x_{12} -18~\\2x_{2} x_{13} -32~\\2x_{6} x_{17} +2x_{3} x_{14} -42~\\2x_{8} x_{19} +2x_{4} x_{15} +2x_{3} x_{14} -48~\\2x_{10} x_{21} +2x_{8} x_{19} +2x_{7} x_{18} +2x_{4} x_{15} -54~\\2x_{9} x_{20} +2x_{7} x_{18} +2x_{5} x_{16} +2x_{4} x_{15} -56~\\2x_{11} x_{22} +4x_{9} x_{20} +2x_{5} x_{16} -58~\\x_{11} x_{22} +2x_{9} x_{20} +x_{5} x_{16} -29~\\\end{array}\)

h-characteristic: (2, 2, 2, 1, 0, 1, 1, 0)
Length of the weight dual to h: 346
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_5+A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+2V_{7\psi}+V_{6\psi}+3V_{4\psi}+2V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 29h_{8}+58h_{7}+56h_{6}+52h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 1/13g_{27}+25/17g_{22}+12/5g_{19}+2/37g_{14}+21/5g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{8} -18~\\2x_{2} x_{9} -32~\\2x_{3} x_{10} -42~\\2x_{4} x_{11} -48~\\2x_{6} x_{13} +2x_{4} x_{11} -52~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{4} x_{11} -56~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} -58~\\2x_{6} x_{13} +x_{5} x_{12} -29~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 0, 0, 0, 2)
Length of the weight dual to h: 336
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_5+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_6+A^{2}_1 Containing regular semisimple subalgebra number 4: C^{1}_5+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+4V_{10\psi}+3V_{8\psi}+V_{6\psi}+7V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 28h_{8}+54h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 1/26g_{44}+1/37g_{34}+4884/5785g_{26}+1603726/3275467g_{22}+360/1157g_{19}+414925/3275467g_{15}+48275/3275467g_{8}+21/5g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{20} +x_{5} x_{19} -x_{4} x_{18} ~\\x_{10} x_{19} +x_{9} x_{15} -x_{8} x_{14} ~\\2x_{1} x_{11} -18~\\2x_{2} x_{12} -32~\\2x_{3} x_{13} -42~\\2x_{8} x_{18} +2x_{4} x_{14} -48~\\2x_{8} x_{18} +2x_{6} x_{16} +2x_{4} x_{14} -50~\\2x_{8} x_{18} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{4} x_{14} -52~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} -54~\\x_{10} x_{20} +2x_{9} x_{19} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} -28~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 336
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2C^{1}_4 Containing regular semisimple subalgebra number 2: A^{2}_7
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+3V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 32h_{8}+62h_{7}+60h_{6}+54h_{5}+48h_{4}+38h_{3}+28h_{2}+14h_{1}\)
\( e = 8/25g_{22}+7g_{16}+3g_{14}+15/37g_{13}+6g_{12}+6/13g_{11}+8/5g_{8}+7/17g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{9} -14~\\2x_{5} x_{13} +2x_{1} x_{9} -28~\\2x_{6} x_{14} +2x_{1} x_{9} -38~\\2x_{6} x_{14} +2x_{2} x_{10} -48~\\2x_{7} x_{15} +2x_{2} x_{10} -54~\\2x_{7} x_{15} +2x_{3} x_{11} -60~\\2x_{8} x_{16} +2x_{3} x_{11} -62~\\x_{8} x_{16} +x_{4} x_{12} -32~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 0, 0, 1, 0)
Length of the weight dual to h: 334
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_6+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_5+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+3V_{10\psi}+2V_{9\psi}+2V_{8\psi}+V_{6\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 27h_{8}+54h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 1/26g_{44}+8298775/18629667g_{34}+3145250/18629667g_{28}+2212859/93148335g_{22}+60000/66773g_{19}+27096/66773g_{12}+21/5g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{5} x_{17} -x_{4} x_{16} ~\\x_{9} x_{17} +x_{8} x_{14} -x_{7} x_{13} ~\\2x_{1} x_{10} -18~\\2x_{2} x_{11} -32~\\2x_{3} x_{12} -42~\\2x_{7} x_{16} +2x_{4} x_{13} -48~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{4} x_{13} -50~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} -52~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} -54~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} -27~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 0, 1, 0, 0)
Length of the weight dual to h: 332
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_6 Containing regular semisimple subalgebra number 2: C^{1}_5+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+2V_{10\psi}+4V_{9\psi}+V_{8\psi}+V_{6\psi}+2V_{2\psi}+4V_{\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 26h_{8}+52h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 1658414/4636227g_{44}+1140161/4636227g_{39}+159539/4636227g_{34}+8772/8933g_{12}+4872/8933g_{4}+21/5g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{16} +x_{5} x_{15} -x_{4} x_{14} ~\\x_{8} x_{15} +x_{7} x_{13} -x_{6} x_{12} ~\\2x_{1} x_{9} -18~\\2x_{2} x_{10} -32~\\2x_{3} x_{11} -42~\\2x_{6} x_{14} +2x_{4} x_{12} -48~\\2x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} -50~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} -52~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} -52~\\x_{8} x_{16} +2x_{7} x_{15} +x_{5} x_{13} -26~\\\end{array}\)

h-characteristic: (2, 2, 2, 2, 1, 0, 0, 0)
Length of the weight dual to h: 330
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_5
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+V_{10\psi}+6V_{9\psi}+V_{6\psi}+V_{2\psi}+21V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 25h_{8}+50h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 25/17g_{44}+12/5g_{4}+21/5g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -18~\\2x_{2} x_{7} -32~\\2x_{3} x_{8} -42~\\2x_{4} x_{9} -48~\\2x_{5} x_{10} -50~\\2x_{5} x_{10} -50~\\2x_{5} x_{10} -50~\\x_{5} x_{10} -25~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 2, 0, 0, 2)
Length of the weight dual to h: 240
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: C^{1}_8 Containing regular semisimple subalgebra number 2: C^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_5+C^{1}_3 Containing regular semisimple subalgebra number 4: 2C^{1}_4 Containing regular semisimple subalgebra number 5: 2C^{1}_4 Containing regular semisimple subalgebra number 6: C^{1}_4+C^{1}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{8\psi}+5V_{6\psi}+2V_{4\psi}+4V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 26h_{8}+50h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\)
\( e = -1862774129104918889557/116384909347372522516030g_{34}-41204379369557065970/1662641562105321750229g_{28}-189780756194684679/1025797293689052535g_{26}+94851924783000888737/1662641562105321750229g_{22}+4122796743095634808111/58192454673686261258015g_{21}+34751815333623107/205159458737810507g_{20}-27510174154811984/3077391881067157605g_{19}-2000084417017/5766202913720g_{17}-10369056482550898736/1662641562105321750229g_{15}+36246433881870742/615478376213431521g_{13}+326524574342420128/3077391881067157605g_{12}+1511344928649/5766202913720g_{11}+177576775263/576620291372g_{10}+98568380939/75871090970g_{9}-380637097129625339706/58192454673686261258015g_{8}-17792357097594449/615478376213431521g_{5}-2943323391/576620291372g_{3}+25442897403/75871090970g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{29} +x_{2} x_{24} -x_{1} x_{23} ~\\x_{13} x_{34} +x_{9} x_{30} -x_{6} x_{29} +x_{3} x_{26} -x_{2} x_{25} ~\\x_{17} x_{36} +x_{15} x_{35} +x_{10} x_{32} -x_{8} x_{30} -x_{3} x_{27} ~\\x_{14} x_{36} +x_{10} x_{35} -x_{8} x_{34} +x_{4} x_{32} -x_{3} x_{31} ~\\x_{14} x_{35} -x_{12} x_{34} +x_{10} x_{33} -x_{9} x_{31} +x_{4} x_{28} ~\\x_{11} x_{25} +x_{6} x_{20} -x_{5} x_{19} ~\\x_{16} x_{31} +x_{12} x_{27} -x_{11} x_{24} +x_{8} x_{21} -x_{7} x_{20} ~\\x_{18} x_{35} +x_{17} x_{33} +x_{14} x_{28} -x_{12} x_{26} -x_{9} x_{21} ~\\x_{17} x_{32} -x_{16} x_{30} +x_{15} x_{28} -x_{13} x_{27} +x_{10} x_{22} ~\\x_{18} x_{32} +x_{17} x_{28} -x_{16} x_{26} +x_{14} x_{22} -x_{13} x_{21} ~\\2x_{5} x_{23} +2x_{1} x_{19} -14~\\2x_{7} x_{25} +2x_{2} x_{20} +2x_{1} x_{19} -24~\\2x_{11} x_{29} +2x_{7} x_{25} +2x_{6} x_{24} +2x_{2} x_{20} -34~\\2x_{13} x_{31} +2x_{9} x_{27} +2x_{6} x_{24} +2x_{3} x_{21} +2x_{2} x_{20} -40~\\2x_{16} x_{34} +2x_{13} x_{31} +2x_{12} x_{30} +2x_{9} x_{27} +2x_{8} x_{26} +2x_{3} x_{21} -46~\\2x_{14} x_{32} +2x_{12} x_{30} +2x_{10} x_{28} +2x_{9} x_{27} +2x_{8} x_{26} +2x_{4} x_{22} +2x_{3} x_{21} -48~\\2x_{17} x_{35} +2x_{15} x_{33} +2x_{14} x_{32} +4x_{10} x_{28} +2x_{8} x_{26} +2x_{4} x_{22} +2x_{3} x_{21} -50~\\x_{18} x_{36} +2x_{17} x_{35} +x_{15} x_{33} +2x_{14} x_{32} +2x_{10} x_{28} +x_{4} x_{22} -26~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 2, 0, 1, 0)
Length of the weight dual to h: 238
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_7 Containing regular semisimple subalgebra number 2: C^{1}_4+C^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+V_{4\psi}+3V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 25h_{8}+50h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\)
\( e = -16355968617/563391847510g_{34}+9377672678/56339184751g_{28}-780018840/56339184751g_{22}-505978834/6345368931g_{19}-4729040279/29227153864g_{17}+14010688063/63453689310g_{13}+1330044715/6345368931g_{12}+7978718343/29227153864g_{11}+21390691113/73067884660g_{10}+2868532793/1922839070g_{9}-131687987/12690737862g_{5}-438346761/73067884660g_{3}+623020041/1922839070g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{24} +x_{2} x_{19} -x_{1} x_{18} ~\\x_{9} x_{25} -x_{6} x_{24} +x_{3} x_{21} -x_{2} x_{20} ~\\x_{10} x_{26} -x_{8} x_{25} +x_{4} x_{23} -x_{3} x_{22} ~\\x_{11} x_{20} +x_{6} x_{15} -x_{5} x_{14} ~\\x_{12} x_{22} -x_{11} x_{19} +x_{8} x_{16} -x_{7} x_{15} ~\\x_{13} x_{23} -x_{12} x_{21} +x_{10} x_{17} -x_{9} x_{16} ~\\2x_{5} x_{18} +2x_{1} x_{14} -14~\\2x_{7} x_{20} +2x_{2} x_{15} +2x_{1} x_{14} -24~\\2x_{11} x_{24} +2x_{7} x_{20} +2x_{6} x_{19} +2x_{2} x_{15} -34~\\2x_{9} x_{22} +2x_{6} x_{19} +2x_{3} x_{16} +2x_{2} x_{15} -40~\\2x_{12} x_{25} +2x_{9} x_{22} +2x_{8} x_{21} +2x_{3} x_{16} -46~\\2x_{10} x_{23} +2x_{8} x_{21} +2x_{4} x_{17} +2x_{3} x_{16} -48~\\2x_{13} x_{26} +4x_{10} x_{23} +2x_{4} x_{17} -50~\\x_{13} x_{26} +2x_{10} x_{23} +x_{4} x_{17} -25~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 1, 1, 0)
Length of the weight dual to h: 226
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{12\psi}+V_{10\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 25h_{8}+50h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\)
\( e = 1/37g_{22}+12/5g_{21}+6/5g_{20}+6g_{16}+5g_{12}+10/17g_{11}+3/13g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{8} -12~\\2x_{6} x_{13} +2x_{1} x_{8} -24~\\2x_{5} x_{12} +2x_{1} x_{8} -32~\\2x_{5} x_{12} +2x_{2} x_{9} -40~\\2x_{4} x_{11} +2x_{2} x_{9} -44~\\2x_{4} x_{11} +2x_{3} x_{10} -48~\\2x_{7} x_{14} +2x_{4} x_{11} +2x_{3} x_{10} -50~\\x_{7} x_{14} +2x_{3} x_{10} -25~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 2, 0, 0)
Length of the weight dual to h: 224
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_6
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{12\psi}+V_{10\psi}+3V_{8\psi}+5V_{6\psi}+3V_{4\psi}+V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 24h_{8}+48h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\)
\( e = 12/5g_{28}+6g_{16}+6/5g_{13}+5g_{12}+10/17g_{11}+3/13g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -12~\\2x_{6} x_{12} +2x_{1} x_{7} -24~\\2x_{5} x_{11} +2x_{1} x_{7} -32~\\2x_{5} x_{11} +2x_{2} x_{8} -40~\\2x_{4} x_{10} +2x_{2} x_{8} -44~\\2x_{4} x_{10} +2x_{3} x_{9} -48~\\4x_{3} x_{9} -48~\\2x_{3} x_{9} -24~\\\end{array}\)

h-characteristic: (2, 2, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 208
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_6+B^{1}_2 Containing regular semisimple subalgebra number 2: C^{1}_4+2B^{1}_2 Containing regular semisimple subalgebra number 3: C^{1}_4+A^{2}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+3V_{10\psi}+2V_{8\psi}+6V_{6\psi}+3V_{4\psi}+4V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 24h_{8}+46h_{7}+44h_{6}+42h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 4/17g_{34}+28055989/63454335g_{26}+19066081/59640140g_{22}+6402833/63454335g_{20}+2088316/12690867g_{19}+3/26g_{18}+72586/69349g_{17}+1243307/11928028g_{15}+6047/12690867g_{13}+18568/69349g_{10}-28475/11928028g_{8}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{23} +x_{3} x_{20} -x_{2} x_{19} ~\\x_{10} x_{24} -x_{8} x_{23} +x_{4} x_{22} -x_{3} x_{21} ~\\x_{11} x_{21} +x_{8} x_{15} -x_{7} x_{14} ~\\x_{12} x_{22} -x_{11} x_{20} +x_{10} x_{16} -x_{9} x_{15} ~\\2x_{1} x_{13} -14~\\2x_{7} x_{19} +2x_{2} x_{14} -24~\\2x_{7} x_{19} +2x_{6} x_{18} +2x_{2} x_{14} -30~\\2x_{9} x_{21} +2x_{6} x_{18} +2x_{3} x_{15} +2x_{2} x_{14} -36~\\2x_{11} x_{23} +2x_{9} x_{21} +2x_{8} x_{20} +2x_{6} x_{18} +2x_{3} x_{15} -42~\\2x_{11} x_{23} +2x_{9} x_{21} +2x_{8} x_{20} +2x_{5} x_{17} +2x_{3} x_{15} -44~\\2x_{10} x_{22} +2x_{8} x_{20} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{3} x_{15} -46~\\x_{12} x_{24} +2x_{10} x_{22} +x_{5} x_{17} +x_{4} x_{16} -24~\\\end{array}\)

h-characteristic: (2, 2, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 192
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: C^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_5+C^{1}_3 Containing regular semisimple subalgebra number 4: C^{1}_5+B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_4+C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 6: C^{1}_4+B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 7: C^{1}_6+A^{2}_1 Containing regular semisimple subalgebra number 8: C^{1}_4+B^{1}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+2V_{10\psi}+3V_{8\psi}+5V_{6\psi}+3V_{4\psi}+7V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+42h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 1/17g_{44}+63782166785065013732217/350677910987464751747000g_{34}-32219274275372145/7156692060968668403g_{31}-7190267555241699437954/43834738873433093968375g_{28}+867140133809038785/7156692060968668403g_{26}+970772469265140159/14313384121937336806g_{25}+11816050409405807264184/43834738873433093968375g_{22}+9393151721702179621531/70135582197492950349400g_{21}+318752514721358001/7156692060968668403g_{19}+703667292052553607/7156692060968668403g_{18}-906986999517433353322/8766947774686618793675g_{15}-144588505401231279/7156692060968668403g_{12}+131590476/154743737g_{10}+589760045547757725393/14027116439498590069880g_{8}+61297842/154743737g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{11} x_{30} +x_{8} x_{26} +x_{3} x_{23} -x_{2} x_{22} ~\\x_{15} x_{32} +x_{13} x_{31} +x_{9} x_{28} -x_{7} x_{26} -x_{3} x_{24} ~\\x_{12} x_{32} +x_{9} x_{31} -x_{7} x_{30} +x_{4} x_{28} -x_{3} x_{27} ~\\x_{12} x_{31} -x_{10} x_{30} +x_{9} x_{29} -x_{8} x_{27} +x_{4} x_{25} ~\\x_{14} x_{27} +x_{10} x_{24} +x_{7} x_{19} -x_{6} x_{18} ~\\x_{16} x_{31} +x_{15} x_{29} +x_{12} x_{25} -x_{10} x_{23} -x_{8} x_{19} ~\\x_{15} x_{28} -x_{14} x_{26} +x_{13} x_{25} -x_{11} x_{24} +x_{9} x_{20} ~\\x_{16} x_{28} +x_{15} x_{25} -x_{14} x_{23} +x_{12} x_{20} -x_{11} x_{19} ~\\2x_{1} x_{17} -14~\\2x_{6} x_{22} +2x_{2} x_{18} -24~\\2x_{11} x_{27} +2x_{8} x_{24} +2x_{3} x_{19} +2x_{2} x_{18} -30~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{3} x_{19} -36~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{3} x_{19} -38~\\2x_{12} x_{28} +2x_{10} x_{26} +2x_{9} x_{25} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} +2x_{3} x_{19} -40~\\2x_{15} x_{31} +2x_{13} x_{29} +2x_{12} x_{28} +4x_{9} x_{25} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} +2x_{3} x_{19} -42~\\x_{16} x_{32} +2x_{15} x_{31} +x_{13} x_{29} +2x_{12} x_{28} +2x_{9} x_{25} +x_{5} x_{21} +x_{4} x_{20} -22~\\\end{array}\)

h-characteristic: (2, 2, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 190
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: C^{1}_7 Containing regular semisimple subalgebra number 2: C^{1}_6+A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_5+B^{1}_2 Containing regular semisimple subalgebra number 4: C^{1}_4+C^{1}_3 Containing regular semisimple subalgebra number 5: C^{1}_4+B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+2V_{10\psi}+2V_{8\psi}+2V_{7\psi}+4V_{6\psi}+2V_{4\psi}+2V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 21h_{8}+42h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 323484083921597990788/2252381358595608690989g_{44}-168233864091493071044/2252381358595608690989g_{39}+1808072966229086670563/13514288151573652145934g_{34}+221249109934748349004/2252381358595608690989g_{33}-196845591742629923920/6757144075786826072967g_{28}-19219703695053051/196766083567363387g_{25}+26516507276069628700/6757144075786826072967g_{22}+3598367177823450/17887825778851217g_{19}+1806007934290704/28109440509623341g_{18}+167406432245160/2555403682693031g_{12}+25661003896088730/196766083567363387g_{11}+53889108/44587253g_{10}-502239848286705/17887825778851217g_{4}+25133460/44587253g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{10} x_{28} +x_{7} x_{24} +x_{3} x_{21} -x_{2} x_{20} ~\\x_{14} x_{30} +x_{12} x_{29} +x_{8} x_{26} -x_{6} x_{24} -x_{3} x_{22} ~\\x_{11} x_{30} +x_{8} x_{29} -x_{6} x_{28} +x_{4} x_{26} -x_{3} x_{25} ~\\x_{11} x_{29} -x_{9} x_{28} +x_{8} x_{27} -x_{7} x_{25} +x_{4} x_{23} ~\\x_{13} x_{25} +x_{9} x_{22} +x_{6} x_{18} -x_{5} x_{17} ~\\x_{15} x_{29} +x_{14} x_{27} +x_{11} x_{23} -x_{9} x_{21} -x_{7} x_{18} ~\\x_{14} x_{26} -x_{13} x_{24} +x_{12} x_{23} -x_{10} x_{22} +x_{8} x_{19} ~\\x_{15} x_{26} +x_{14} x_{23} -x_{13} x_{21} +x_{11} x_{19} -x_{10} x_{18} ~\\2x_{1} x_{16} -14~\\2x_{5} x_{20} +2x_{2} x_{17} -24~\\2x_{10} x_{25} +2x_{7} x_{22} +2x_{3} x_{18} +2x_{2} x_{17} -30~\\2x_{13} x_{28} +2x_{10} x_{25} +2x_{9} x_{24} +2x_{7} x_{22} +2x_{6} x_{21} +2x_{3} x_{18} -36~\\2x_{11} x_{26} +2x_{9} x_{24} +2x_{8} x_{23} +2x_{7} x_{22} +2x_{6} x_{21} +2x_{4} x_{19} +2x_{3} x_{18} -38~\\2x_{14} x_{29} +2x_{12} x_{27} +2x_{11} x_{26} +4x_{8} x_{23} +2x_{6} x_{21} +2x_{4} x_{19} +2x_{3} x_{18} -40~\\2x_{15} x_{30} +4x_{14} x_{29} +2x_{12} x_{27} +4x_{11} x_{26} +4x_{8} x_{23} +2x_{4} x_{19} -42~\\x_{15} x_{30} +2x_{14} x_{29} +x_{12} x_{27} +2x_{11} x_{26} +2x_{8} x_{23} +x_{4} x_{19} -21~\\\end{array}\)

h-characteristic: (2, 2, 0, 2, 0, 1, 0, 0)
Length of the weight dual to h: 188
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_6 Containing regular semisimple subalgebra number 2: C^{1}_4+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+2V_{10\psi}+V_{8\psi}+4V_{7\psi}+3V_{6\psi}+V_{4\psi}+4V_{3\psi}+2V_{2\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 20h_{8}+40h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 45381202/232288455g_{44}+8806142/46457691g_{39}-514912/46457691g_{34}+28271343/108401279g_{18}+339552705/1409216627g_{12}+31316406/108401279g_{11}+3326076/2212271g_{10}-5589225/1409216627g_{4}+1170300/2212271g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{19} +x_{3} x_{16} -x_{2} x_{15} ~\\x_{8} x_{20} -x_{6} x_{19} +x_{4} x_{18} -x_{3} x_{17} ~\\x_{9} x_{17} +x_{6} x_{13} -x_{5} x_{12} ~\\x_{10} x_{18} -x_{9} x_{16} +x_{8} x_{14} -x_{7} x_{13} ~\\2x_{1} x_{11} -14~\\2x_{5} x_{15} +2x_{2} x_{12} -24~\\2x_{7} x_{17} +2x_{3} x_{13} +2x_{2} x_{12} -30~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{3} x_{13} -36~\\2x_{8} x_{18} +2x_{6} x_{16} +2x_{4} x_{14} +2x_{3} x_{13} -38~\\2x_{10} x_{20} +4x_{8} x_{18} +2x_{4} x_{14} -40~\\2x_{10} x_{20} +4x_{8} x_{18} +2x_{4} x_{14} -40~\\x_{10} x_{20} +2x_{8} x_{18} +x_{4} x_{14} -20~\\\end{array}\)

h-characteristic: (2, 2, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 186
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_5+A^{2}_2 Containing regular semisimple subalgebra number 2: C^{1}_4+A^{2}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+2V_{9\psi}+V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+3V_{2\psi}
+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 21h_{8}+42h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 1431671/2502310g_{34}+2/17g_{32}+27298/250231g_{28}+6525/6763g_{25}+7112/1251155g_{22}+1/13g_{20}+1860/6763g_{18}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{4} x_{17} -x_{3} x_{16} ~\\x_{9} x_{17} +x_{8} x_{13} -x_{7} x_{12} ~\\2x_{1} x_{10} -14~\\2x_{2} x_{11} -24~\\2x_{7} x_{16} +2x_{3} x_{12} -30~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} -34~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{3} x_{12} -38~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -40~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} -42~\\x_{9} x_{18} +2x_{8} x_{17} +2x_{5} x_{14} +x_{4} x_{13} -21~\\\end{array}\)

h-characteristic: (2, 2, 1, 0, 1, 1, 0, 0)
Length of the weight dual to h: 184
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_4+A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+2V_{9\psi}+4V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+6V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 20h_{8}+40h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 2/17g_{38}+8/5g_{34}+3g_{18}+1/13g_{13}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -14~\\2x_{2} x_{8} -24~\\2x_{3} x_{9} -30~\\2x_{5} x_{11} +2x_{3} x_{9} -34~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{3} x_{9} -38~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} -40~\\4x_{5} x_{11} +2x_{4} x_{10} -40~\\2x_{5} x_{11} +x_{4} x_{10} -20~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 176
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: C^{1}_5+3A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_4+4A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_5+A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 4: C^{1}_4+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_4+2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+4V_{8\psi}+5V_{6\psi}+11V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 20h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 1/17g_{52}+1/26g_{44}+1/37g_{34}+366/397g_{31}+165/794g_{25}+818696/1351785g_{22}+22186/270357g_{15}+1817/540714g_{8}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{20} +x_{4} x_{19} -x_{3} x_{18} ~\\x_{10} x_{19} +x_{9} x_{14} -x_{8} x_{13} ~\\2x_{1} x_{11} -14~\\2x_{2} x_{12} -24~\\2x_{8} x_{18} +2x_{3} x_{13} -30~\\2x_{8} x_{18} +2x_{5} x_{15} +2x_{3} x_{13} -32~\\2x_{8} x_{18} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{3} x_{13} -34~\\2x_{8} x_{18} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{3} x_{13} -36~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} -38~\\x_{10} x_{20} +2x_{9} x_{19} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} -20~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 174
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_4+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_5+A^{2}_1 Containing regular semisimple subalgebra number 4: C^{1}_4+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+3V_{8\psi}+2V_{7\psi}+4V_{6\psi}+7V_{2\psi}+6V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 19h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 1/17g_{52}+1/26g_{44}+1431671/2502310g_{34}+27298/250231g_{28}+6525/6763g_{25}+7112/1251155g_{22}+1860/6763g_{18}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{4} x_{17} -x_{3} x_{16} ~\\x_{9} x_{17} +x_{8} x_{13} -x_{7} x_{12} ~\\2x_{1} x_{10} -14~\\2x_{2} x_{11} -24~\\2x_{7} x_{16} +2x_{3} x_{12} -30~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} -32~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{3} x_{12} -34~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -36~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} -38~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} -19~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 172
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_5+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_4+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+2V_{8\psi}+4V_{7\psi}+3V_{6\psi}+4V_{2\psi}+8V_{\psi}+11V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+36h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 1/17g_{52}+793016/1532047g_{44}+234686/1532047g_{39}+29915/3064094g_{34}+1818/1763g_{18}+1335/3526g_{11}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{16} +x_{4} x_{15} -x_{3} x_{14} ~\\x_{8} x_{15} +x_{7} x_{12} -x_{6} x_{11} ~\\2x_{1} x_{9} -14~\\2x_{2} x_{10} -24~\\2x_{6} x_{14} +2x_{3} x_{11} -30~\\2x_{6} x_{14} +2x_{5} x_{13} +2x_{3} x_{11} -32~\\2x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} -34~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} -36~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} -36~\\x_{8} x_{16} +2x_{7} x_{15} +x_{5} x_{13} +x_{4} x_{12} -18~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 1, 0, 0, 0)
Length of the weight dual to h: 170
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_5 Containing regular semisimple subalgebra number 2: C^{1}_4+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+V_{8\psi}+6V_{7\psi}+2V_{6\psi}+2V_{2\psi}+6V_{\psi}+21V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 17h_{8}+34h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 841/2074g_{52}+746/3111g_{48}+40/3111g_{44}+209/183g_{11}+100/183g_{3}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{4} x_{13} -x_{3} x_{12} ~\\x_{7} x_{13} +x_{6} x_{11} -x_{5} x_{10} ~\\2x_{1} x_{8} -14~\\2x_{2} x_{9} -24~\\2x_{5} x_{12} +2x_{3} x_{10} -30~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} -32~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} -34~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} -34~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} -34~\\x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} -17~\\\end{array}\)

h-characteristic: (2, 2, 2, 1, 0, 0, 0, 0)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_4
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+8V_{7\psi}+V_{6\psi}+V_{2\psi}+36V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+32h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 8/5g_{52}+3g_{3}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -14~\\2x_{2} x_{6} -24~\\2x_{3} x_{7} -30~\\2x_{4} x_{8} -32~\\2x_{4} x_{8} -32~\\2x_{4} x_{8} -32~\\2x_{4} x_{8} -32~\\x_{4} x_{8} -16~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 160
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_5+C^{1}_3 Containing regular semisimple subalgebra number 2: 2C^{1}_3+B^{1}_2 Containing regular semisimple subalgebra number 3: A^{2}_5+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{10\psi}+3V_{8\psi}+6V_{6\psi}+3V_{4\psi}+6V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+42h_{7}+40h_{6}+38h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\)
\( e = 9/26g_{34}+2746640861/7132156424g_{26}+11102915/14039678g_{23}+2095451713/7153215941g_{22}+549685335/7132156424g_{20}+353259145/3566078212g_{19}+8/17g_{18}+1597175/14039678g_{16}+515687922/7153215941g_{15}-2215429/3566078212g_{13}-16871070/7153215941g_{8}+1/2g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{23} +x_{2} x_{20} -x_{1} x_{19} ~\\x_{10} x_{24} -x_{8} x_{23} +x_{3} x_{22} -x_{2} x_{21} ~\\x_{11} x_{21} +x_{8} x_{14} -x_{7} x_{13} ~\\x_{12} x_{22} -x_{11} x_{20} +x_{10} x_{15} -x_{9} x_{14} ~\\2x_{7} x_{19} +2x_{1} x_{13} -10~\\2x_{7} x_{19} +2x_{4} x_{16} +2x_{1} x_{13} -20~\\2x_{7} x_{19} +2x_{5} x_{17} +2x_{1} x_{13} -26~\\2x_{9} x_{21} +2x_{5} x_{17} +2x_{2} x_{14} +2x_{1} x_{13} -32~\\2x_{11} x_{23} +2x_{9} x_{21} +2x_{8} x_{20} +2x_{5} x_{17} +2x_{2} x_{14} -38~\\2x_{11} x_{23} +2x_{9} x_{21} +2x_{8} x_{20} +2x_{6} x_{18} +2x_{2} x_{14} -40~\\2x_{10} x_{22} +2x_{8} x_{20} +2x_{6} x_{18} +2x_{3} x_{15} +2x_{2} x_{14} -42~\\x_{12} x_{24} +2x_{10} x_{22} +x_{6} x_{18} +x_{3} x_{15} -22~\\\end{array}\)

h-characteristic: (1, 0, 1, 1, 0, 1, 1, 0)
Length of the weight dual to h: 150
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_4+C^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}
+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 21h_{8}+42h_{7}+40h_{6}+36h_{5}+32h_{4}+26h_{3}+18h_{2}+10h_{1}\)
\( e = 4g_{24}+9/37g_{22}+3g_{21}+6/5g_{20}+4/13g_{19}+5/17g_{16}+2/5g_{11}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{5} x_{12} -10~\\2x_{5} x_{12} +2x_{1} x_{8} -18~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{1} x_{8} -26~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{1} x_{8} -32~\\2x_{6} x_{13} +2x_{3} x_{10} +2x_{1} x_{8} -36~\\2x_{6} x_{13} +2x_{3} x_{10} +2x_{2} x_{9} -40~\\2x_{7} x_{14} +2x_{3} x_{10} +2x_{2} x_{9} -42~\\x_{7} x_{14} +2x_{2} x_{9} -21~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 144
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: 2C^{1}_4 Containing regular semisimple subalgebra number 2: C^{1}_4+C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 3: 2C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 4: A^{2}_5+2A^{1}_1 Containing regular semisimple subalgebra number 5: 2C^{1}_3+A^{2}_1 Containing regular semisimple subalgebra number 6: A^{2}_5+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{10\psi}+V_{8\psi}+7V_{6\psi}+5V_{4\psi}+6V_{2\psi}+2V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 20h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\)
\( e = 557261/8049424g_{44}+403879/8049424g_{39}-1891/8049424g_{34}+288/257g_{26}+119121/170905g_{22}+40/257g_{19}+2468/13597g_{18}+5g_{16}+2238/34181g_{15}+1028/13597g_{11}-11/34181g_{8}+1/2g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{23} +x_{3} x_{20} -x_{2} x_{19} ~\\x_{10} x_{24} +x_{6} x_{22} -x_{5} x_{21} ~\\x_{11} x_{20} +x_{8} x_{15} -x_{7} x_{14} ~\\x_{12} x_{22} +x_{10} x_{18} -x_{9} x_{17} ~\\2x_{1} x_{13} -10~\\2x_{4} x_{16} +2x_{1} x_{13} -20~\\2x_{9} x_{21} +2x_{5} x_{17} +2x_{1} x_{13} -26~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{5} x_{17} +2x_{2} x_{14} -32~\\2x_{10} x_{22} +2x_{7} x_{19} +2x_{6} x_{18} +2x_{5} x_{17} +2x_{2} x_{14} -34~\\2x_{12} x_{24} +4x_{10} x_{22} +2x_{7} x_{19} +2x_{6} x_{18} +2x_{2} x_{14} -36~\\2x_{12} x_{24} +4x_{10} x_{22} +2x_{8} x_{20} +2x_{6} x_{18} +2x_{3} x_{15} +2x_{2} x_{14} -38~\\x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{8} x_{20} +x_{6} x_{18} +x_{3} x_{15} -20~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 142
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_4+C^{1}_3 Containing regular semisimple subalgebra number 2: 2C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_5+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{10\psi}+V_{8\psi}+5V_{6\psi}+4V_{5\psi}+3V_{4\psi}+4V_{2\psi}+2V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 19h_{8}+38h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\)
\( e = 9/26g_{44}+101679/140215g_{34}+8946/140215g_{28}-13/140215g_{22}+1056/967g_{19}+5g_{16}+152/967g_{12}+8/17g_{11}+1/2g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{3} x_{17} -x_{2} x_{16} ~\\x_{9} x_{17} +x_{8} x_{12} -x_{7} x_{11} ~\\2x_{1} x_{10} -10~\\2x_{4} x_{13} +2x_{1} x_{10} -20~\\2x_{5} x_{14} +2x_{1} x_{10} -26~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{2} x_{11} -32~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{2} x_{11} -34~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{3} x_{12} +2x_{2} x_{11} -36~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{3} x_{12} -38~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +x_{3} x_{12} -19~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 1, 0, 0)
Length of the weight dual to h: 140
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2C^{1}_3 Containing regular semisimple subalgebra number 2: A^{2}_5
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{10\psi}+V_{8\psi}+3V_{6\psi}+8V_{5\psi}+V_{4\psi}+3V_{2\psi}+11V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+36h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\)
\( e = 9/26g_{44}+9/5g_{34}+5g_{16}+4g_{12}+8/17g_{11}+1/2g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -10~\\2x_{4} x_{10} +2x_{1} x_{7} -20~\\2x_{5} x_{11} +2x_{1} x_{7} -26~\\2x_{5} x_{11} +2x_{2} x_{8} -32~\\2x_{6} x_{12} +2x_{2} x_{8} -34~\\2x_{6} x_{12} +2x_{3} x_{9} -36~\\2x_{6} x_{12} +2x_{3} x_{9} -36~\\x_{6} x_{12} +x_{3} x_{9} -18~\\\end{array}\)

h-characteristic: (2, 0, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 112
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 9
Containing regular semisimple subalgebra number 1: C^{1}_6+B^{1}_2 Containing regular semisimple subalgebra number 2: C^{1}_5+C^{1}_3 Containing regular semisimple subalgebra number 3: C^{1}_5+B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 4: C^{1}_4+2B^{1}_2 Containing regular semisimple subalgebra number 5: 2C^{1}_3+B^{1}_2 Containing regular semisimple subalgebra number 6: 2C^{1}_3+B^{1}_2 Containing regular semisimple subalgebra number 7: C^{1}_3+2B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 8: C^{1}_4+A^{2}_3 Containing regular semisimple subalgebra number 9: A^{2}_3+C^{1}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{8\psi}+7V_{6\psi}+6V_{4\psi}+9V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 2/5g_{44}+132242607481928330814108/392050076370340160716835g_{34}-1338722597333851144/20848182737056110647g_{31}-192374849655012675623108/392050076370340160716835g_{28}+1976200387078963192/20848182737056110647g_{26}+289832542138850334/20848182737056110647g_{25}+862724827071219988660994/1176150229111020482150505g_{22}+22601080490218100948426/78410015274068032143367g_{21}+754805565718223826/20848182737056110647g_{19}+4350879557118554297/62544548211168331941g_{18}+3/17g_{17}+189261715/464662857g_{16}-83346757034589613280981/235230045822204096430101g_{15}-1129010369502322067/62544548211168331941g_{12}+82078945/464662857g_{9}+40607709999466102316245/235230045822204096430101g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{11} x_{30} +x_{8} x_{26} +x_{2} x_{23} -x_{1} x_{22} ~\\x_{15} x_{32} +x_{13} x_{31} +x_{9} x_{28} -x_{7} x_{26} -x_{2} x_{24} ~\\x_{12} x_{32} +x_{9} x_{31} -x_{7} x_{30} +x_{3} x_{28} -x_{2} x_{27} ~\\x_{12} x_{31} -x_{10} x_{30} +x_{9} x_{29} -x_{8} x_{27} +x_{3} x_{25} ~\\x_{14} x_{27} +x_{10} x_{24} +x_{7} x_{18} -x_{6} x_{17} ~\\x_{16} x_{31} +x_{15} x_{29} +x_{12} x_{25} -x_{10} x_{23} -x_{8} x_{18} ~\\x_{15} x_{28} -x_{14} x_{26} +x_{13} x_{25} -x_{11} x_{24} +x_{9} x_{19} ~\\x_{16} x_{28} +x_{15} x_{25} -x_{14} x_{23} +x_{12} x_{19} -x_{11} x_{18} ~\\2x_{6} x_{22} +2x_{1} x_{17} -10~\\2x_{6} x_{22} +2x_{5} x_{21} +2x_{1} x_{17} -16~\\2x_{11} x_{27} +2x_{8} x_{24} +2x_{5} x_{21} +2x_{2} x_{18} +2x_{1} x_{17} -22~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{2} x_{18} -28~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{4} x_{20} +2x_{2} x_{18} -30~\\2x_{12} x_{28} +2x_{10} x_{26} +2x_{9} x_{25} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{4} x_{20} +2x_{3} x_{19} +2x_{2} x_{18} -32~\\2x_{15} x_{31} +2x_{13} x_{29} +2x_{12} x_{28} +4x_{9} x_{25} +2x_{7} x_{23} +2x_{4} x_{20} +2x_{3} x_{19} +2x_{2} x_{18} -34~\\x_{16} x_{32} +2x_{15} x_{31} +x_{13} x_{29} +2x_{12} x_{28} +2x_{9} x_{25} +x_{4} x_{20} +x_{3} x_{19} -18~\\\end{array}\)

h-characteristic: (2, 0, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 110
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_5+B^{1}_2 Containing regular semisimple subalgebra number 2: C^{1}_3+2B^{1}_2 Containing regular semisimple subalgebra number 3: A^{2}_3+C^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{8\psi}+6V_{6\psi}+2V_{5\psi}+3V_{4\psi}+4V_{3\psi}+6V_{2\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 17h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 2/5g_{44}+6220869119/24495624330g_{34}+456800093/4899124866g_{28}+2529943449/7401366061g_{25}-18972851/4899124866g_{22}+797325220/7401366061g_{19}+1902053797/14802732122g_{18}+3/17g_{17}+23486465/26339381g_{16}-8502906/7401366061g_{12}+4161940/26339381g_{9}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{21} +x_{2} x_{18} -x_{1} x_{17} ~\\x_{9} x_{22} -x_{7} x_{21} +x_{3} x_{20} -x_{2} x_{19} ~\\x_{10} x_{19} +x_{7} x_{13} -x_{6} x_{12} ~\\x_{11} x_{20} -x_{10} x_{18} +x_{9} x_{14} -x_{8} x_{13} ~\\2x_{6} x_{17} +2x_{1} x_{12} -10~\\2x_{6} x_{17} +2x_{5} x_{16} +2x_{1} x_{12} -16~\\2x_{8} x_{19} +2x_{5} x_{16} +2x_{2} x_{13} +2x_{1} x_{12} -22~\\2x_{10} x_{21} +2x_{8} x_{19} +2x_{7} x_{18} +2x_{5} x_{16} +2x_{2} x_{13} -28~\\2x_{10} x_{21} +2x_{8} x_{19} +2x_{7} x_{18} +2x_{4} x_{15} +2x_{2} x_{13} -30~\\2x_{9} x_{20} +2x_{7} x_{18} +2x_{4} x_{15} +2x_{3} x_{14} +2x_{2} x_{13} -32~\\2x_{11} x_{22} +4x_{9} x_{20} +2x_{4} x_{15} +2x_{3} x_{14} -34~\\x_{11} x_{22} +2x_{9} x_{20} +x_{4} x_{15} +x_{3} x_{14} -17~\\\end{array}\)

h-characteristic: (2, 0, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 106
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_5+A^{2}_2 Containing regular semisimple subalgebra number 2: C^{1}_3+A^{2}_2+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+2V_{7\psi}+3V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 17h_{8}+34h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 6220869119/24495624330g_{34}+1/5g_{32}+2529943449/7401366061g_{30}+456800093/4899124866g_{28}+797325220/7401366061g_{25}+1902053797/14802732122g_{24}-18972851/4899124866g_{22}+2/17g_{20}-8502906/7401366061g_{18}+23486465/26339381g_{9}+4161940/26339381g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{21} +x_{2} x_{18} -x_{1} x_{17} ~\\x_{9} x_{22} -x_{7} x_{21} +x_{3} x_{20} -x_{2} x_{19} ~\\x_{10} x_{19} +x_{7} x_{13} -x_{6} x_{12} ~\\x_{11} x_{20} -x_{10} x_{18} +x_{9} x_{14} -x_{8} x_{13} ~\\2x_{6} x_{17} +2x_{1} x_{12} -10~\\2x_{8} x_{19} +2x_{2} x_{13} +2x_{1} x_{12} -16~\\2x_{10} x_{21} +2x_{8} x_{19} +2x_{7} x_{18} +2x_{2} x_{13} -22~\\2x_{10} x_{21} +2x_{8} x_{19} +2x_{7} x_{18} +2x_{4} x_{15} +2x_{2} x_{13} -26~\\2x_{10} x_{21} +2x_{8} x_{19} +2x_{7} x_{18} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{2} x_{13} -30~\\2x_{9} x_{20} +2x_{7} x_{18} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{3} x_{14} +2x_{2} x_{13} -32~\\2x_{11} x_{22} +4x_{9} x_{20} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{3} x_{14} -34~\\x_{11} x_{22} +2x_{9} x_{20} +2x_{4} x_{15} +x_{3} x_{14} -17~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 102
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_4+C^{1}_3 Containing regular semisimple subalgebra number 2: A^{2}_4+B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 17h_{8}+34h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\)
\( e = -95477/2111022g_{34}+82138/1055511g_{28}+3g_{27}+6/5g_{26}+985/13031g_{25}+4g_{23}-130388/5277555g_{22}+604/13031g_{18}+2/5g_{10}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{6} x_{17} -x_{5} x_{16} ~\\x_{9} x_{17} +x_{8} x_{15} -x_{7} x_{14} ~\\2x_{1} x_{10} -8~\\2x_{4} x_{13} +2x_{1} x_{10} -16~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{1} x_{10} -22~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{1} x_{10} -26~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{2} x_{11} -30~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{2} x_{11} -32~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{3} x_{12} +2x_{2} x_{11} -34~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +2x_{2} x_{11} -17~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 1, 1, 0, 0)
Length of the weight dual to h: 100
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_4+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+7V_{4\psi}+4V_{3\psi}+2V_{2\psi}+2V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+32h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\)
\( e = 4/17g_{34}+3g_{33}+4g_{23}+6/5g_{19}+3/26g_{18}+2/5g_{10}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -8~\\2x_{4} x_{10} +2x_{1} x_{7} -16~\\2x_{6} x_{12} +2x_{4} x_{10} +2x_{1} x_{7} -22~\\2x_{6} x_{12} +2x_{3} x_{9} +2x_{1} x_{7} -26~\\2x_{6} x_{12} +2x_{3} x_{9} +2x_{2} x_{8} -30~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -32~\\2x_{5} x_{11} +4x_{2} x_{8} -32~\\x_{5} x_{11} +2x_{2} x_{8} -16~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 96
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 13
Containing regular semisimple subalgebra number 1: C^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_5+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 4: C^{1}_4+B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 5: 2C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 6: 2C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 7: C^{1}_3+B^{1}_2+3A^{1}_1 Containing regular semisimple subalgebra number 8: C^{1}_6+A^{2}_1 Containing regular semisimple subalgebra number 9: C^{1}_5+A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 10: C^{1}_4+B^{1}_2+A^{2}_1 Containing regular semisimple subalgebra number 11: 2C^{1}_3+A^{2}_1 Containing regular semisimple subalgebra number 12: 2C^{1}_3+A^{2}_1 Containing regular semisimple subalgebra number 13: C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+6V_{6\psi}+7V_{4\psi}+12V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{52}+1/17g_{44}-1338722597333851144/20848182737056110647g_{36}+132242607481928330814108/392050076370340160716835g_{34}+1976200387078963192/20848182737056110647g_{31}+289832542138850334/20848182737056110647g_{30}-192374849655012675623108/392050076370340160716835g_{28}+754805565718223826/20848182737056110647g_{25}+4350879557118554297/62544548211168331941g_{24}+862724827071219988660994/1176150229111020482150505g_{22}+22601080490218100948426/78410015274068032143367g_{21}-1129010369502322067/62544548211168331941g_{18}-83346757034589613280981/235230045822204096430101g_{15}+189261715/464662857g_{9}+40607709999466102316245/235230045822204096430101g_{8}+82078945/464662857g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{11} x_{30} +x_{8} x_{26} +x_{2} x_{23} -x_{1} x_{22} ~\\x_{15} x_{32} +x_{13} x_{31} +x_{9} x_{28} -x_{7} x_{26} -x_{2} x_{24} ~\\x_{12} x_{32} +x_{9} x_{31} -x_{7} x_{30} +x_{3} x_{28} -x_{2} x_{27} ~\\x_{12} x_{31} -x_{10} x_{30} +x_{9} x_{29} -x_{8} x_{27} +x_{3} x_{25} ~\\x_{14} x_{27} +x_{10} x_{24} +x_{7} x_{18} -x_{6} x_{17} ~\\x_{16} x_{31} +x_{15} x_{29} +x_{12} x_{25} -x_{10} x_{23} -x_{8} x_{18} ~\\x_{15} x_{28} -x_{14} x_{26} +x_{13} x_{25} -x_{11} x_{24} +x_{9} x_{19} ~\\x_{16} x_{28} +x_{15} x_{25} -x_{14} x_{23} +x_{12} x_{19} -x_{11} x_{18} ~\\2x_{6} x_{22} +2x_{1} x_{17} -10~\\2x_{11} x_{27} +2x_{8} x_{24} +2x_{2} x_{18} +2x_{1} x_{17} -16~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{2} x_{18} -22~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{4} x_{20} +2x_{2} x_{18} -24~\\2x_{14} x_{30} +2x_{11} x_{27} +2x_{10} x_{26} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} +2x_{2} x_{18} -26~\\2x_{12} x_{28} +2x_{10} x_{26} +2x_{9} x_{25} +2x_{8} x_{24} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} +2x_{3} x_{19} +2x_{2} x_{18} -28~\\2x_{15} x_{31} +2x_{13} x_{29} +2x_{12} x_{28} +4x_{9} x_{25} +2x_{7} x_{23} +2x_{5} x_{21} +2x_{4} x_{20} +2x_{3} x_{19} +2x_{2} x_{18} -30~\\x_{16} x_{32} +2x_{15} x_{31} +x_{13} x_{29} +2x_{12} x_{28} +2x_{9} x_{25} +x_{5} x_{21} +x_{4} x_{20} +x_{3} x_{19} -16~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 96
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_4+A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+5V_{6\psi}+10V_{4\psi}+6V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+32h_{7}+32h_{6}+28h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\)
\( e = 2/17g_{32}+4g_{29}+3g_{28}+6/5g_{25}+1/13g_{20}+2/5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -8~\\2x_{4} x_{10} +2x_{1} x_{7} -16~\\2x_{3} x_{9} +2x_{1} x_{7} -20~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{1} x_{7} -24~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{3} x_{9} +2x_{1} x_{7} -28~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -32~\\2x_{6} x_{12} +2x_{5} x_{11} +4x_{2} x_{8} -32~\\2x_{5} x_{11} +2x_{2} x_{8} -16~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 94
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 9
Containing regular semisimple subalgebra number 1: C^{1}_6+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+C^{1}_3 Containing regular semisimple subalgebra number 4: C^{1}_4+B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 5: 2C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 6: 2C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 7: C^{1}_3+B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 8: C^{1}_5+A^{2}_1 Containing regular semisimple subalgebra number 9: C^{1}_3+B^{1}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+5V_{6\psi}+2V_{5\psi}+5V_{4\psi}+2V_{3\psi}+8V_{2\psi}+4V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{52}+91570690136186865025/696837167445421471392g_{44}-3017479376755215931/19356587984595040872g_{39}+341370651012169815/1613048998716253406g_{34}+83716797463633637495/696837167445421471392g_{33}-80733628649129150/659883681293012757g_{30}-1531102894186465757/19356587984595040872g_{28}+93160667387239876/659883681293012757g_{25}+2094763109988763/219961227097670919g_{24}+20300000995516201969/696837167445421471392g_{22}+10913992231055626/219961227097670919g_{18}+58626966588564574/659883681293012757g_{17}-14966538283267124/659883681293012757g_{11}+128051795/210928277g_{9}+54505270/210928277g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{10} x_{28} +x_{7} x_{24} +x_{2} x_{21} -x_{1} x_{20} ~\\x_{14} x_{30} +x_{12} x_{29} +x_{8} x_{26} -x_{6} x_{24} -x_{2} x_{22} ~\\x_{11} x_{30} +x_{8} x_{29} -x_{6} x_{28} +x_{3} x_{26} -x_{2} x_{25} ~\\x_{11} x_{29} -x_{9} x_{28} +x_{8} x_{27} -x_{7} x_{25} +x_{3} x_{23} ~\\x_{13} x_{25} +x_{9} x_{22} +x_{6} x_{17} -x_{5} x_{16} ~\\x_{15} x_{29} +x_{14} x_{27} +x_{11} x_{23} -x_{9} x_{21} -x_{7} x_{17} ~\\x_{14} x_{26} -x_{13} x_{24} +x_{12} x_{23} -x_{10} x_{22} +x_{8} x_{18} ~\\x_{15} x_{26} +x_{14} x_{23} -x_{13} x_{21} +x_{11} x_{18} -x_{10} x_{17} ~\\2x_{5} x_{20} +2x_{1} x_{16} -10~\\2x_{10} x_{25} +2x_{7} x_{22} +2x_{2} x_{17} +2x_{1} x_{16} -16~\\2x_{13} x_{28} +2x_{10} x_{25} +2x_{9} x_{24} +2x_{7} x_{22} +2x_{6} x_{21} +2x_{2} x_{17} -22~\\2x_{13} x_{28} +2x_{10} x_{25} +2x_{9} x_{24} +2x_{7} x_{22} +2x_{6} x_{21} +2x_{4} x_{19} +2x_{2} x_{17} -24~\\2x_{11} x_{26} +2x_{9} x_{24} +2x_{8} x_{23} +2x_{7} x_{22} +2x_{6} x_{21} +2x_{4} x_{19} +2x_{3} x_{18} +2x_{2} x_{17} -26~\\2x_{14} x_{29} +2x_{12} x_{27} +2x_{11} x_{26} +4x_{8} x_{23} +2x_{6} x_{21} +2x_{4} x_{19} +2x_{3} x_{18} +2x_{2} x_{17} -28~\\2x_{15} x_{30} +4x_{14} x_{29} +2x_{12} x_{27} +4x_{11} x_{26} +4x_{8} x_{23} +2x_{4} x_{19} +2x_{3} x_{18} -30~\\x_{15} x_{30} +2x_{14} x_{29} +x_{12} x_{27} +2x_{11} x_{26} +2x_{8} x_{23} +x_{4} x_{19} +x_{3} x_{18} -15~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 92
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: C^{1}_6 Containing regular semisimple subalgebra number 2: C^{1}_5+A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+B^{1}_2 Containing regular semisimple subalgebra number 4: 2C^{1}_3 Containing regular semisimple subalgebra number 5: 2C^{1}_3 Containing regular semisimple subalgebra number 6: C^{1}_3+B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+4V_{6\psi}+4V_{5\psi}+3V_{4\psi}+4V_{3\psi}+5V_{2\psi}+4V_{\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 47744816154731154292/471829422582185238537g_{52}-44881111686734087252/471829422582185238537g_{48}+57360678748712363770/471829422582185238537g_{44}+46527830960734610614/471829422582185238537g_{43}-13981985061354277013/471829422582185238537g_{39}+1875723157691620453/471829422582185238537g_{34}-2470915324531696/9203015907901173g_{24}+2166543174990608/9203015907901173g_{18}-73861238625454/9203015907901173g_{17}+675677993583854/9203015907901173g_{11}+1101764408825981/9203015907901173g_{10}+6208005/6439871g_{9}-279790347438391/9203015907901173g_{3}+2599135/6439871g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{26} +x_{6} x_{22} +x_{2} x_{19} -x_{1} x_{18} ~\\x_{13} x_{28} +x_{11} x_{27} +x_{7} x_{24} -x_{5} x_{22} -x_{2} x_{20} ~\\x_{10} x_{28} +x_{7} x_{27} -x_{5} x_{26} +x_{3} x_{24} -x_{2} x_{23} ~\\x_{10} x_{27} -x_{8} x_{26} +x_{7} x_{25} -x_{6} x_{23} +x_{3} x_{21} ~\\x_{12} x_{23} +x_{8} x_{20} +x_{5} x_{16} -x_{4} x_{15} ~\\x_{14} x_{27} +x_{13} x_{25} +x_{10} x_{21} -x_{8} x_{19} -x_{6} x_{16} ~\\x_{13} x_{24} -x_{12} x_{22} +x_{11} x_{21} -x_{9} x_{20} +x_{7} x_{17} ~\\x_{14} x_{24} +x_{13} x_{21} -x_{12} x_{19} +x_{10} x_{17} -x_{9} x_{16} ~\\2x_{4} x_{18} +2x_{1} x_{15} -10~\\2x_{9} x_{23} +2x_{6} x_{20} +2x_{2} x_{16} +2x_{1} x_{15} -16~\\2x_{12} x_{26} +2x_{9} x_{23} +2x_{8} x_{22} +2x_{6} x_{20} +2x_{5} x_{19} +2x_{2} x_{16} -22~\\2x_{10} x_{24} +2x_{8} x_{22} +2x_{7} x_{21} +2x_{6} x_{20} +2x_{5} x_{19} +2x_{3} x_{17} +2x_{2} x_{16} -24~\\2x_{13} x_{27} +2x_{11} x_{25} +2x_{10} x_{24} +4x_{7} x_{21} +2x_{5} x_{19} +2x_{3} x_{17} +2x_{2} x_{16} -26~\\2x_{14} x_{28} +4x_{13} x_{27} +2x_{11} x_{25} +4x_{10} x_{24} +4x_{7} x_{21} +2x_{3} x_{17} -28~\\2x_{14} x_{28} +4x_{13} x_{27} +2x_{11} x_{25} +4x_{10} x_{24} +4x_{7} x_{21} +2x_{3} x_{17} -28~\\x_{14} x_{28} +2x_{13} x_{27} +x_{11} x_{25} +2x_{10} x_{24} +2x_{7} x_{21} +x_{3} x_{17} -14~\\\end{array}\)

h-characteristic: (2, 1, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_4+A^{2}_2+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+A^{2}_2+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+5V_{4\psi}+6V_{3\psi}+5V_{2\psi}+4V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\)
\( e = 1/26g_{44}+1/5g_{37}+101679/140215g_{34}+1056/967g_{30}+8946/140215g_{28}+2/17g_{26}+152/967g_{24}-13/140215g_{22}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{3} x_{17} -x_{2} x_{16} ~\\x_{9} x_{17} +x_{8} x_{12} -x_{7} x_{11} ~\\2x_{1} x_{10} -10~\\2x_{7} x_{16} +2x_{2} x_{11} -16~\\2x_{7} x_{16} +2x_{4} x_{13} +2x_{2} x_{11} -20~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{2} x_{11} -24~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{2} x_{11} -26~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} -28~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -30~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +2x_{4} x_{13} +x_{3} x_{12} -15~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 1, 0, 0, 0)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_5 Containing regular semisimple subalgebra number 2: C^{1}_3+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+3V_{6\psi}+6V_{5\psi}+V_{4\psi}+6V_{3\psi}+3V_{2\psi}+21V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 738295/8225937g_{52}+1498759/8225937g_{48}-117884/8225937g_{44}+30275/598823g_{17}+497596/1796469g_{11}+301843/1197646g_{10}+42425/31517g_{9}-13030/1796469g_{3}+11516/31517g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{17} +x_{2} x_{14} -x_{1} x_{13} ~\\x_{7} x_{18} -x_{5} x_{17} +x_{3} x_{16} -x_{2} x_{15} ~\\x_{8} x_{15} +x_{5} x_{11} -x_{4} x_{10} ~\\x_{9} x_{16} -x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} ~\\2x_{4} x_{13} +2x_{1} x_{10} -10~\\2x_{6} x_{15} +2x_{2} x_{11} +2x_{1} x_{10} -16~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{2} x_{11} -22~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{2} x_{11} -24~\\2x_{9} x_{18} +4x_{7} x_{16} +2x_{3} x_{12} -26~\\2x_{9} x_{18} +4x_{7} x_{16} +2x_{3} x_{12} -26~\\2x_{9} x_{18} +4x_{7} x_{16} +2x_{3} x_{12} -26~\\x_{9} x_{18} +2x_{7} x_{16} +x_{3} x_{12} -13~\\\end{array}\)

h-characteristic: (2, 1, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 88
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_4+A^{2}_2 Containing regular semisimple subalgebra number 2: C^{1}_3+A^{2}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{7\psi}+2V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\)
\( e = 157782/230887g_{44}+1/5g_{42}+20565/230887g_{39}-37/230887g_{34}+708/619g_{24}+2/17g_{19}+136/619g_{17}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{16} +x_{3} x_{15} -x_{2} x_{14} ~\\x_{8} x_{15} +x_{7} x_{11} -x_{6} x_{10} ~\\2x_{1} x_{9} -10~\\2x_{6} x_{14} +2x_{2} x_{10} -16~\\2x_{6} x_{14} +2x_{4} x_{12} +2x_{2} x_{10} -20~\\2x_{6} x_{14} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{2} x_{10} -24~\\2x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} -26~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} -28~\\2x_{8} x_{16} +4x_{7} x_{15} +4x_{4} x_{12} +2x_{3} x_{11} -28~\\x_{8} x_{16} +2x_{7} x_{15} +2x_{4} x_{12} +x_{3} x_{11} -14~\\\end{array}\)

h-characteristic: (2, 1, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 86
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{7\psi}+V_{6\psi}+6V_{5\psi}+3V_{4\psi}+2V_{3\psi}+10V_{2\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\)
\( e = 1/5g_{47}+9/5g_{44}+4g_{17}+2/17g_{12}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -10~\\2x_{2} x_{7} -16~\\2x_{4} x_{9} +2x_{2} x_{7} -20~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{2} x_{7} -24~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} -26~\\4x_{4} x_{9} +2x_{3} x_{8} -26~\\4x_{4} x_{9} +2x_{3} x_{8} -26~\\2x_{4} x_{9} +x_{3} x_{8} -13~\\\end{array}\)

h-characteristic: (0, 2, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 86
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_4+3A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_4+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+V_{6\psi}+6V_{5\psi}+3V_{4\psi}+6V_{3\psi}+7V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 15h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\)
\( e = 1/17g_{44}+1/26g_{34}+3g_{32}+6/5g_{31}+1/37g_{22}+4g_{16}+2/5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{8} -8~\\2x_{4} x_{11} +2x_{1} x_{8} -16~\\2x_{3} x_{10} +2x_{1} x_{8} -20~\\2x_{3} x_{10} +2x_{2} x_{9} -24~\\2x_{5} x_{12} +2x_{3} x_{10} +2x_{2} x_{9} -26~\\2x_{6} x_{13} +2x_{5} x_{12} +2x_{3} x_{10} +2x_{2} x_{9} -28~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +2x_{3} x_{10} +2x_{2} x_{9} -30~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +2x_{2} x_{9} -15~\\\end{array}\)

h-characteristic: (0, 2, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 84
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_4+2A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_4+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+V_{6\psi}+4V_{5\psi}+7V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+7V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+28h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\)
\( e = 1/17g_{44}+3g_{38}+1/26g_{34}+6/5g_{25}+4g_{16}+2/5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -8~\\2x_{4} x_{10} +2x_{1} x_{7} -16~\\2x_{3} x_{9} +2x_{1} x_{7} -20~\\2x_{3} x_{9} +2x_{2} x_{8} -24~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -26~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -28~\\2x_{6} x_{12} +2x_{5} x_{11} +4x_{2} x_{8} -28~\\x_{6} x_{12} +x_{5} x_{11} +2x_{2} x_{8} -14~\\\end{array}\)

h-characteristic: (0, 2, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 82
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_4+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+V_{6\psi}+2V_{5\psi}+11V_{4\psi}+2V_{3\psi}+2V_{2\psi}+4V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{8}+26h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\)
\( e = 1/17g_{44}+3g_{43}+6/5g_{18}+4g_{16}+2/5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -8~\\2x_{4} x_{9} +2x_{1} x_{6} -16~\\2x_{3} x_{8} +2x_{1} x_{6} -20~\\2x_{3} x_{8} +2x_{2} x_{7} -24~\\2x_{5} x_{10} +2x_{3} x_{8} +2x_{2} x_{7} -26~\\2x_{5} x_{10} +4x_{2} x_{7} -26~\\2x_{5} x_{10} +4x_{2} x_{7} -26~\\x_{5} x_{10} +2x_{2} x_{7} -13~\\\end{array}\)

h-characteristic: (2, 2, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: C^{1}_4+4A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+5A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 4: C^{1}_3+A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_4+2A^{2}_1 Containing regular semisimple subalgebra number 6: C^{1}_3+2A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+6V_{6\psi}+5V_{4\psi}+16V_{2\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{58}+1/17g_{52}+1/26g_{44}+6828/6457g_{36}+1/37g_{34}+760/6457g_{30}+18545526/24633455g_{22}+236541/4926691g_{15}-353/4926691g_{8}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{20} +x_{3} x_{19} -x_{2} x_{18} ~\\x_{10} x_{19} +x_{9} x_{13} -x_{8} x_{12} ~\\2x_{1} x_{11} -10~\\2x_{8} x_{18} +2x_{2} x_{12} -16~\\2x_{8} x_{18} +2x_{4} x_{14} +2x_{2} x_{12} -18~\\2x_{8} x_{18} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{2} x_{12} -20~\\2x_{8} x_{18} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{2} x_{12} -22~\\2x_{8} x_{18} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{2} x_{12} -24~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{2} x_{12} -26~\\x_{10} x_{20} +2x_{9} x_{19} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +x_{3} x_{13} -14~\\\end{array}\)

h-characteristic: (0, 0, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 4B^{1}_2 Containing regular semisimple subalgebra number 2: A^{2}_3+2B^{1}_2 Containing regular semisimple subalgebra number 3: 2A^{2}_3
sl(2)-module decomposition of the ambient Lie algebra: \(10V_{6\psi}+6V_{4\psi}+10V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+30h_{7}+28h_{6}+26h_{5}+24h_{4}+18h_{3}+12h_{2}+6h_{1}\)
\( e = 4g_{44}+4/5g_{34}+3/50g_{26}+3/26g_{25}+3/10g_{24}+3/2g_{23}+4/17g_{22}+4/37g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{10} -6~\\2x_{4} x_{12} +2x_{2} x_{10} -12~\\2x_{6} x_{14} +2x_{4} x_{12} +2x_{2} x_{10} -18~\\2x_{8} x_{16} +2x_{6} x_{14} +2x_{4} x_{12} +2x_{2} x_{10} -24~\\2x_{8} x_{16} +2x_{6} x_{14} +2x_{4} x_{12} +2x_{1} x_{9} -26~\\2x_{8} x_{16} +2x_{6} x_{14} +2x_{3} x_{11} +2x_{1} x_{9} -28~\\2x_{8} x_{16} +2x_{5} x_{13} +2x_{3} x_{11} +2x_{1} x_{9} -30~\\x_{7} x_{15} +x_{5} x_{13} +x_{3} x_{11} +x_{1} x_{9} -16~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_4
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+V_{6\psi}+15V_{4\psi}+V_{2\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\)
\( e = 3g_{48}+4g_{16}+6/5g_{11}+2/5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -8~\\2x_{4} x_{8} +2x_{1} x_{5} -16~\\2x_{3} x_{7} +2x_{1} x_{5} -20~\\2x_{3} x_{7} +2x_{2} x_{6} -24~\\4x_{2} x_{6} -24~\\4x_{2} x_{6} -24~\\4x_{2} x_{6} -24~\\2x_{2} x_{6} -12~\\\end{array}\)

h-characteristic: (2, 2, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 78
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: C^{1}_4+3A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+4A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 4: C^{1}_3+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_3+2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+5V_{6\psi}+2V_{5\psi}+4V_{4\psi}+11V_{2\psi}+8V_{\psi}+9V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{58}+1/17g_{52}+1/26g_{44}+101679/140215g_{34}+1056/967g_{30}+8946/140215g_{28}+152/967g_{24}-13/140215g_{22}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{3} x_{17} -x_{2} x_{16} ~\\x_{9} x_{17} +x_{8} x_{12} -x_{7} x_{11} ~\\2x_{1} x_{10} -10~\\2x_{7} x_{16} +2x_{2} x_{11} -16~\\2x_{7} x_{16} +2x_{4} x_{13} +2x_{2} x_{11} -18~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{2} x_{11} -20~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{2} x_{11} -22~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} -24~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -26~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} -13~\\\end{array}\)

h-characteristic: (2, 2, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 76
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_4+2A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_4+A^{2}_1 Containing regular semisimple subalgebra number 4: C^{1}_3+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+4V_{6\psi}+4V_{5\psi}+3V_{4\psi}+7V_{2\psi}+12V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{58}+1/17g_{52}+157782/230887g_{44}+20565/230887g_{39}-37/230887g_{34}+708/619g_{24}+136/619g_{17}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{16} +x_{3} x_{15} -x_{2} x_{14} ~\\x_{8} x_{15} +x_{7} x_{11} -x_{6} x_{10} ~\\2x_{1} x_{9} -10~\\2x_{6} x_{14} +2x_{2} x_{10} -16~\\2x_{6} x_{14} +2x_{4} x_{12} +2x_{2} x_{10} -18~\\2x_{6} x_{14} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{2} x_{10} -20~\\2x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} -22~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} -24~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} -24~\\x_{8} x_{16} +2x_{7} x_{15} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} -12~\\\end{array}\)

h-characteristic: (2, 2, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 74
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_4+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+3V_{6\psi}+6V_{5\psi}+2V_{4\psi}+4V_{2\psi}+12V_{\psi}+22V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 11h_{8}+22h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{58}+1010835/1650251g_{52}+220938/1650251g_{48}-1093/1650251g_{44}+4128/3361g_{17}+1096/3361g_{10}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{3} x_{13} -x_{2} x_{12} ~\\x_{7} x_{13} +x_{6} x_{10} -x_{5} x_{9} ~\\2x_{1} x_{8} -10~\\2x_{5} x_{12} +2x_{2} x_{9} -16~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{2} x_{9} -18~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} -20~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} -22~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} -22~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} -22~\\x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} -11~\\\end{array}\)

h-characteristic: (2, 2, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_4 Containing regular semisimple subalgebra number 2: C^{1}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{6\psi}+8V_{5\psi}+V_{4\psi}+2V_{2\psi}+8V_{\psi}+36V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 2606/5671g_{58}+1313/5671g_{55}-37/5671g_{52}+148/107g_{10}+56/107g_{2}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{12} +x_{3} x_{11} -x_{2} x_{10} ~\\x_{6} x_{11} +x_{5} x_{9} -x_{4} x_{8} ~\\2x_{1} x_{7} -10~\\2x_{4} x_{10} +2x_{2} x_{8} -16~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -18~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} -20~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} -20~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} -20~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} -20~\\x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} -10~\\\end{array}\)

h-characteristic: (2, 2, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 70
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{6\psi}+10V_{5\psi}+V_{2\psi}+55V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 9/5g_{58}+4g_{2}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{4} -10~\\2x_{2} x_{5} -16~\\2x_{3} x_{6} -18~\\2x_{3} x_{6} -18~\\2x_{3} x_{6} -18~\\2x_{3} x_{6} -18~\\2x_{3} x_{6} -18~\\x_{3} x_{6} -9~\\\end{array}\)

h-characteristic: (0, 0, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 64
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: 2C^{1}_3+B^{1}_2 Containing regular semisimple subalgebra number 2: C^{1}_3+2B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 3: 3B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 4: A^{2}_3+C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 5: A^{2}_3+B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 6: 3B^{1}_2+A^{2}_1 Containing regular semisimple subalgebra number 7: A^{2}_3+B^{1}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(6V_{6\psi}+9V_{4\psi}+15V_{2\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\)
\( e = 4/17g_{52}+144335/1356736g_{44}+5853/6539g_{40}+33629/1356736g_{39}+372/6539g_{35}-1433/1356736g_{34}+891/4930g_{24}+8777447/15026622g_{22}+159/4930g_{17}+291014/7513311g_{15}-3092/7513311g_{8}+3/26g_{3}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{23} +x_{2} x_{20} -x_{1} x_{19} ~\\x_{10} x_{24} +x_{4} x_{22} -x_{3} x_{21} ~\\x_{11} x_{20} +x_{8} x_{14} -x_{7} x_{13} ~\\x_{12} x_{22} +x_{10} x_{16} -x_{9} x_{15} ~\\2x_{7} x_{19} +2x_{1} x_{13} -6~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{3} x_{15} +2x_{1} x_{13} -12~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{6} x_{18} +2x_{3} x_{15} +2x_{1} x_{13} -18~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{5} x_{17} +2x_{3} x_{15} +2x_{1} x_{13} -20~\\2x_{10} x_{22} +2x_{7} x_{19} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{3} x_{15} +2x_{1} x_{13} -22~\\2x_{12} x_{24} +4x_{10} x_{22} +2x_{7} x_{19} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{1} x_{13} -24~\\2x_{12} x_{24} +4x_{10} x_{22} +2x_{8} x_{20} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{2} x_{14} +2x_{1} x_{13} -26~\\x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{8} x_{20} +x_{5} x_{17} +x_{4} x_{16} +x_{2} x_{14} -14~\\\end{array}\)

h-characteristic: (0, 0, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 62
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_3+2B^{1}_2 Containing regular semisimple subalgebra number 2: 3B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_3+C^{1}_3 Containing regular semisimple subalgebra number 4: A^{2}_3+B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(6V_{6\psi}+6V_{4\psi}+6V_{3\psi}+10V_{2\psi}+2V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\)
\( e = 4/5g_{52}+4/17g_{44}+5745/6503g_{35}+1862179/3082422g_{34}+372/6503g_{29}+58822/1541211g_{28}-2948/7706055g_{22}+3/26g_{11}+3/10g_{10}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{2} x_{17} -x_{1} x_{16} ~\\x_{9} x_{17} +x_{8} x_{11} -x_{7} x_{10} ~\\2x_{7} x_{16} +2x_{1} x_{10} -6~\\2x_{7} x_{16} +2x_{4} x_{13} +2x_{1} x_{10} -12~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{4} x_{13} +2x_{1} x_{10} -18~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{3} x_{12} +2x_{1} x_{10} -20~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{1} x_{10} -22~\\2x_{8} x_{17} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{2} x_{11} +2x_{1} x_{10} -24~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{2} x_{11} -26~\\x_{9} x_{18} +2x_{8} x_{17} +x_{5} x_{14} +x_{3} x_{12} +x_{2} x_{11} -13~\\\end{array}\)

h-characteristic: (0, 0, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 3B^{1}_2 Containing regular semisimple subalgebra number 2: A^{2}_3+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(6V_{6\psi}+3V_{4\psi}+12V_{3\psi}+6V_{2\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\)
\( e = 4g_{52}+4/5g_{44}+4/17g_{34}+3/26g_{18}+3/10g_{17}+3/2g_{16}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{8} -6~\\2x_{4} x_{10} +2x_{2} x_{8} -12~\\2x_{6} x_{12} +2x_{4} x_{10} +2x_{2} x_{8} -18~\\2x_{6} x_{12} +2x_{4} x_{10} +2x_{1} x_{7} -20~\\2x_{6} x_{12} +2x_{3} x_{9} +2x_{1} x_{7} -22~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{1} x_{7} -24~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{1} x_{7} -24~\\x_{5} x_{11} +x_{3} x_{9} +x_{1} x_{7} -12~\\\end{array}\)

h-characteristic: (0, 1, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 58
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{2}_2+B^{1}_2 Containing regular semisimple subalgebra number 2: A^{2}_2+2B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_3+A^{2}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+4V_{5\psi}+6V_{4\psi}+6V_{3\psi}+7V_{2\psi}+6V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\)
\( e = 4/17g_{44}+2/5g_{37}+5745/6503g_{35}+1862179/3082422g_{34}+372/6503g_{29}+58822/1541211g_{28}+1/5g_{26}-2948/7706055g_{22}+3/26g_{17}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{2} x_{17} -x_{1} x_{16} ~\\x_{9} x_{17} +x_{8} x_{11} -x_{7} x_{10} ~\\2x_{7} x_{16} +2x_{1} x_{10} -6~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{1} x_{10} -12~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{3} x_{12} +2x_{1} x_{10} -16~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -20~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -22~\\2x_{8} x_{17} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} +2x_{1} x_{10} -24~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} -26~\\x_{9} x_{18} +2x_{8} x_{17} +x_{5} x_{14} +2x_{3} x_{12} +x_{2} x_{11} -13~\\\end{array}\)

h-characteristic: (0, 1, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 56
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_2+2B^{1}_2 Containing regular semisimple subalgebra number 2: A^{2}_3+A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+4V_{5\psi}+4V_{4\psi}+8V_{3\psi}+8V_{2\psi}+4V_{\psi}+7V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\)
\( e = 4/5g_{44}+2g_{42}+4/17g_{34}+3/26g_{24}+3/10g_{23}+g_{19}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{4} x_{10} -6~\\2x_{6} x_{12} +2x_{4} x_{10} -12~\\2x_{6} x_{12} +2x_{4} x_{10} +2x_{1} x_{7} -16~\\2x_{6} x_{12} +2x_{4} x_{10} +2x_{2} x_{8} +2x_{1} x_{7} -20~\\2x_{6} x_{12} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -22~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -24~\\2x_{5} x_{11} +2x_{3} x_{9} +4x_{1} x_{7} -24~\\x_{5} x_{11} +x_{3} x_{9} +2x_{1} x_{7} -12~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 1, 1, 0, 0)
Length of the weight dual to h: 52
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A^{2}_2+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+4V_{5\psi}+10V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+24h_{6}+22h_{5}+18h_{4}+14h_{3}+10h_{2}+6h_{1}\)
\( e = 2g_{46}+4/17g_{34}+2/5g_{32}+3/26g_{29}+g_{25}+1/5g_{20}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{6} x_{12} -6~\\2x_{6} x_{12} +2x_{1} x_{7} -10~\\2x_{6} x_{12} +2x_{2} x_{8} +2x_{1} x_{7} -14~\\2x_{6} x_{12} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -18~\\2x_{6} x_{12} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -22~\\2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -24~\\2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +4x_{1} x_{7} -24~\\x_{5} x_{11} +2x_{3} x_{9} +2x_{1} x_{7} -12~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 48
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 10
Containing regular semisimple subalgebra number 1: 2C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+B^{1}_2+3A^{1}_1 Containing regular semisimple subalgebra number 3: 2B^{1}_2+4A^{1}_1 Containing regular semisimple subalgebra number 4: A^{2}_3+4A^{1}_1 Containing regular semisimple subalgebra number 5: 2C^{1}_3+A^{2}_1 Containing regular semisimple subalgebra number 6: C^{1}_3+B^{1}_2+A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 7: 2B^{1}_2+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 8: A^{2}_3+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 9: 2B^{1}_2+2A^{2}_1 Containing regular semisimple subalgebra number 10: A^{2}_3+2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+9V_{4\psi}+21V_{2\psi}+7V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 1/17g_{58}+1/26g_{52}+144335/1356736g_{44}+5853/6539g_{40}+33629/1356736g_{39}+372/6539g_{35}-1433/1356736g_{34}+891/4930g_{24}+8777447/15026622g_{22}+159/4930g_{17}+291014/7513311g_{15}-3092/7513311g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{23} +x_{2} x_{20} -x_{1} x_{19} ~\\x_{10} x_{24} +x_{4} x_{22} -x_{3} x_{21} ~\\x_{11} x_{20} +x_{8} x_{14} -x_{7} x_{13} ~\\x_{12} x_{22} +x_{10} x_{16} -x_{9} x_{15} ~\\2x_{7} x_{19} +2x_{1} x_{13} -6~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{3} x_{15} +2x_{1} x_{13} -12~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{5} x_{17} +2x_{3} x_{15} +2x_{1} x_{13} -14~\\2x_{9} x_{21} +2x_{7} x_{19} +2x_{6} x_{18} +2x_{5} x_{17} +2x_{3} x_{15} +2x_{1} x_{13} -16~\\2x_{10} x_{22} +2x_{7} x_{19} +2x_{6} x_{18} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{3} x_{15} +2x_{1} x_{13} -18~\\2x_{12} x_{24} +4x_{10} x_{22} +2x_{7} x_{19} +2x_{6} x_{18} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{1} x_{13} -20~\\2x_{12} x_{24} +4x_{10} x_{22} +2x_{8} x_{20} +2x_{6} x_{18} +2x_{5} x_{17} +2x_{4} x_{16} +2x_{2} x_{14} +2x_{1} x_{13} -22~\\x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{8} x_{20} +x_{6} x_{18} +x_{5} x_{17} +x_{4} x_{16} +x_{2} x_{14} -12~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 46
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: 2C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 3: 2B^{1}_2+3A^{1}_1 Containing regular semisimple subalgebra number 4: A^{2}_3+3A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_3+B^{1}_2+A^{2}_1 Containing regular semisimple subalgebra number 6: 2B^{1}_2+A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 7: A^{2}_3+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+7V_{4\psi}+4V_{3\psi}+15V_{2\psi}+6V_{\psi}+7V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 11h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 1/17g_{58}+721999/7719215g_{52}+50503/1543843g_{48}-2873/1543843g_{44}+1566/1679g_{35}+1084192/2023195g_{34}+267/3358g_{29}+109118/2023195g_{28}-3101/4046390g_{22}+2253/12005g_{17}+99/2401g_{10}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{21} +x_{2} x_{18} -x_{1} x_{17} ~\\x_{9} x_{22} +x_{4} x_{20} -x_{3} x_{19} ~\\x_{10} x_{18} +x_{7} x_{13} -x_{6} x_{12} ~\\x_{11} x_{20} +x_{9} x_{15} -x_{8} x_{14} ~\\2x_{6} x_{17} +2x_{1} x_{12} -6~\\2x_{8} x_{19} +2x_{6} x_{17} +2x_{3} x_{14} +2x_{1} x_{12} -12~\\2x_{8} x_{19} +2x_{6} x_{17} +2x_{5} x_{16} +2x_{3} x_{14} +2x_{1} x_{12} -14~\\2x_{9} x_{20} +2x_{6} x_{17} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{3} x_{14} +2x_{1} x_{12} -16~\\2x_{11} x_{22} +4x_{9} x_{20} +2x_{6} x_{17} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{1} x_{12} -18~\\2x_{11} x_{22} +4x_{9} x_{20} +2x_{7} x_{18} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{2} x_{13} +2x_{1} x_{12} -20~\\2x_{11} x_{22} +2x_{10} x_{21} +4x_{9} x_{20} +4x_{7} x_{18} +2x_{5} x_{16} +2x_{4} x_{15} +2x_{2} x_{13} -22~\\x_{11} x_{22} +x_{10} x_{21} +2x_{9} x_{20} +2x_{7} x_{18} +x_{5} x_{16} +x_{4} x_{15} +x_{2} x_{13} -11~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 44
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: 2C^{1}_3 Containing regular semisimple subalgebra number 2: C^{1}_3+B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 3: 2B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 4: A^{2}_3+2A^{1}_1 Containing regular semisimple subalgebra number 5: 2B^{1}_2+A^{2}_1 Containing regular semisimple subalgebra number 6: A^{2}_3+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+5V_{4\psi}+8V_{3\psi}+10V_{2\psi}+8V_{\psi}+12V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 51151/719040g_{58}+6611/143808g_{55}-541/143808g_{52}+9593/21378g_{44}+866/10689g_{39}-236/138957g_{34}+507/509g_{29}+60/509g_{23}+339/1712g_{10}+93/1712g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{19} +x_{2} x_{16} -x_{1} x_{15} ~\\x_{8} x_{20} +x_{4} x_{18} -x_{3} x_{17} ~\\x_{9} x_{16} +x_{6} x_{12} -x_{5} x_{11} ~\\x_{10} x_{18} +x_{8} x_{14} -x_{7} x_{13} ~\\2x_{5} x_{15} +2x_{1} x_{11} -6~\\2x_{7} x_{17} +2x_{5} x_{15} +2x_{3} x_{13} +2x_{1} x_{11} -12~\\2x_{8} x_{18} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{1} x_{11} -14~\\2x_{10} x_{20} +4x_{8} x_{18} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{1} x_{11} -16~\\2x_{10} x_{20} +4x_{8} x_{18} +2x_{6} x_{16} +2x_{4} x_{14} +2x_{2} x_{12} +2x_{1} x_{11} -18~\\2x_{10} x_{20} +2x_{9} x_{19} +4x_{8} x_{18} +4x_{6} x_{16} +2x_{4} x_{14} +2x_{2} x_{12} -20~\\2x_{10} x_{20} +2x_{9} x_{19} +4x_{8} x_{18} +4x_{6} x_{16} +2x_{4} x_{14} +2x_{2} x_{12} -20~\\x_{10} x_{20} +x_{9} x_{19} +2x_{8} x_{18} +2x_{6} x_{16} +x_{4} x_{14} +x_{2} x_{12} -10~\\\end{array}\)

h-characteristic: (1, 0, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{2}_2+2A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+B^{1}_2+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_2+A^{2}_1 Containing regular semisimple subalgebra number 4: A^{2}_2+B^{1}_2+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{5\psi}+6V_{4\psi}+8V_{3\psi}+11V_{2\psi}+8V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 11h_{8}+22h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\)
\( e = 1/17g_{52}+1/26g_{44}+2/5g_{41}+5745/6503g_{35}+1862179/3082422g_{34}+1/5g_{31}+372/6503g_{29}+58822/1541211g_{28}-2948/7706055g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{2} x_{17} -x_{1} x_{16} ~\\x_{9} x_{17} +x_{8} x_{11} -x_{7} x_{10} ~\\2x_{7} x_{16} +2x_{1} x_{10} -6~\\2x_{7} x_{16} +2x_{3} x_{12} +2x_{1} x_{10} -10~\\2x_{7} x_{16} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -14~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -16~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -18~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} +2x_{1} x_{10} -20~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} -22~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +2x_{3} x_{12} +x_{2} x_{11} -11~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_3+B^{1}_2 Containing regular semisimple subalgebra number 2: 2B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+3V_{4\psi}+12V_{3\psi}+6V_{2\psi}+6V_{\psi}+22V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 4/5g_{58}+466471/902398g_{52}+34838/451199g_{48}-596/451199g_{44}+1437/1499g_{23}+180/1499g_{16}+3/10g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{13} -x_{1} x_{12} ~\\x_{7} x_{13} +x_{6} x_{9} -x_{5} x_{8} ~\\2x_{5} x_{12} +2x_{1} x_{8} -6~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{1} x_{8} -12~\\2x_{5} x_{12} +2x_{3} x_{10} +2x_{1} x_{8} -14~\\2x_{6} x_{13} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -16~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{3} x_{10} +2x_{2} x_{9} -18~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{3} x_{10} +2x_{2} x_{9} -18~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{3} x_{10} +2x_{2} x_{9} -18~\\x_{7} x_{14} +2x_{6} x_{13} +x_{3} x_{10} +x_{2} x_{9} -9~\\\end{array}\)

h-characteristic: (1, 0, 1, 0, 0, 1, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{2}_2+A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_2+B^{1}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{5\psi}+5V_{4\psi}+8V_{3\psi}+11V_{2\psi}+10V_{\psi}+7V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\)
\( e = 1/17g_{52}+2/5g_{46}+1203616/2110347g_{44}+111022/2110347g_{39}-2845/4220694g_{34}+1518/1663g_{29}+1/5g_{25}+267/3326g_{23}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{16} +x_{2} x_{15} -x_{1} x_{14} ~\\x_{8} x_{15} +x_{7} x_{10} -x_{6} x_{9} ~\\2x_{6} x_{14} +2x_{1} x_{9} -6~\\2x_{6} x_{14} +2x_{3} x_{11} +2x_{1} x_{9} -10~\\2x_{6} x_{14} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{1} x_{9} -14~\\2x_{6} x_{14} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{1} x_{9} -16~\\2x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -18~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} -20~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +4x_{3} x_{11} +2x_{2} x_{10} -20~\\x_{8} x_{16} +2x_{7} x_{15} +x_{5} x_{13} +2x_{3} x_{11} +x_{2} x_{10} -10~\\\end{array}\)

h-characteristic: (0, 2, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2B^{1}_2 Containing regular semisimple subalgebra number 2: A^{2}_3
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+V_{4\psi}+16V_{3\psi}+3V_{2\psi}+37V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 4g_{58}+4/5g_{52}+3/10g_{10}+3/2g_{9}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{6} -6~\\2x_{4} x_{8} +2x_{2} x_{6} -12~\\2x_{4} x_{8} +2x_{1} x_{5} -14~\\2x_{3} x_{7} +2x_{1} x_{5} -16~\\2x_{3} x_{7} +2x_{1} x_{5} -16~\\2x_{3} x_{7} +2x_{1} x_{5} -16~\\2x_{3} x_{7} +2x_{1} x_{5} -16~\\x_{3} x_{7} +x_{1} x_{5} -8~\\\end{array}\)

h-characteristic: (1, 0, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 38
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{2}_2 Containing regular semisimple subalgebra number 2: A^{2}_2+B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{5\psi}+4V_{4\psi}+8V_{3\psi}+12V_{2\psi}+8V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\)
\( e = 466471/902398g_{52}+2/5g_{50}+34838/451199g_{48}-596/451199g_{44}+1437/1499g_{23}+1/5g_{18}+180/1499g_{16}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{13} -x_{1} x_{12} ~\\x_{7} x_{13} +x_{6} x_{9} -x_{5} x_{8} ~\\2x_{5} x_{12} +2x_{1} x_{8} -6~\\2x_{5} x_{12} +2x_{3} x_{10} +2x_{1} x_{8} -10~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{1} x_{8} -14~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -16~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} -18~\\2x_{7} x_{14} +4x_{6} x_{13} +4x_{3} x_{10} +2x_{2} x_{9} -18~\\2x_{7} x_{14} +4x_{6} x_{13} +4x_{3} x_{10} +2x_{2} x_{9} -18~\\x_{7} x_{14} +2x_{6} x_{13} +2x_{3} x_{10} +x_{2} x_{9} -9~\\\end{array}\)

h-characteristic: (1, 0, 1, 1, 0, 0, 0, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_2+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{5\psi}+3V_{4\psi}+8V_{3\psi}+14V_{2\psi}+2V_{\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\)
\( e = 2g_{54}+4/5g_{52}+3/10g_{16}+g_{11}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{4} x_{8} -6~\\2x_{4} x_{8} +2x_{1} x_{5} -10~\\2x_{4} x_{8} +2x_{2} x_{6} +2x_{1} x_{5} -14~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -16~\\2x_{3} x_{7} +4x_{1} x_{5} -16~\\2x_{3} x_{7} +4x_{1} x_{5} -16~\\2x_{3} x_{7} +4x_{1} x_{5} -16~\\x_{3} x_{7} +2x_{1} x_{5} -8~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2A^{2}_2+2A^{1}_1 Containing regular semisimple subalgebra number 2: 2A^{2}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(10V_{4\psi}+8V_{3\psi}+9V_{2\psi}+8V_{\psi}+11V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{49}+1/17g_{44}+2/5g_{37}+1/26g_{34}+g_{30}+1/5g_{26}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -4~\\2x_{2} x_{8} +2x_{1} x_{7} -8~\\2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -12~\\2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -16~\\2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -18~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -20~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +4x_{1} x_{7} -20~\\x_{6} x_{12} +x_{5} x_{11} +2x_{3} x_{9} +2x_{1} x_{7} -10~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 34
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A^{2}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(10V_{4\psi}+4V_{3\psi}+15V_{2\psi}+6V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{53}+1/17g_{44}+2/5g_{42}+g_{24}+1/5g_{19}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -4~\\2x_{2} x_{7} +2x_{1} x_{6} -8~\\2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -12~\\2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -16~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -18~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +4x_{1} x_{6} -18~\\2x_{5} x_{10} +4x_{3} x_{8} +4x_{1} x_{6} -18~\\x_{5} x_{10} +2x_{3} x_{8} +2x_{1} x_{6} -9~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: C^{1}_3+5A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+6A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 4: B^{1}_2+A^{2}_1+4A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_3+2A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 6: B^{1}_2+2A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 7: B^{1}_2+3A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+6V_{4\psi}+28V_{2\psi}+15V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{62}+1/10g_{58}+1/17g_{52}+1/26g_{44}+4998/5791g_{40}+495/11582g_{35}+1/37g_{34}+14753888/23517251g_{22}+680990/23517251g_{15}-10981/47034502g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{9} x_{20} +x_{2} x_{19} -x_{1} x_{18} ~\\x_{10} x_{19} +x_{9} x_{12} -x_{8} x_{11} ~\\2x_{8} x_{18} +2x_{1} x_{11} -6~\\2x_{8} x_{18} +2x_{3} x_{13} +2x_{1} x_{11} -8~\\2x_{8} x_{18} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{1} x_{11} -10~\\2x_{8} x_{18} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{1} x_{11} -12~\\2x_{8} x_{18} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{1} x_{11} -14~\\2x_{8} x_{18} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{1} x_{11} -16~\\2x_{9} x_{19} +2x_{7} x_{17} +2x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +2x_{3} x_{13} +2x_{2} x_{12} +2x_{1} x_{11} -18~\\x_{10} x_{20} +2x_{9} x_{19} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +x_{3} x_{13} +x_{2} x_{12} -10~\\\end{array}\)

h-characteristic: (0, 0, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(10V_{4\psi}+22V_{2\psi}+20V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{56}+2/5g_{47}+g_{17}+1/5g_{12}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -4~\\2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -12~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -16~\\2x_{4} x_{8} +2x_{3} x_{7} +4x_{1} x_{5} -16~\\4x_{3} x_{7} +4x_{1} x_{5} -16~\\4x_{3} x_{7} +4x_{1} x_{5} -16~\\2x_{3} x_{7} +2x_{1} x_{5} -8~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 30
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1: C^{1}_3+4A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+5A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 4: B^{1}_2+A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 5: C^{1}_3+2A^{2}_1 Containing regular semisimple subalgebra number 6: B^{1}_2+2A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+5V_{4\psi}+2V_{3\psi}+21V_{2\psi}+10V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{62}+1/10g_{58}+1/17g_{52}+1/26g_{44}+5745/6503g_{35}+1862179/3082422g_{34}+372/6503g_{29}+58822/1541211g_{28}-2948/7706055g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{18} +x_{2} x_{17} -x_{1} x_{16} ~\\x_{9} x_{17} +x_{8} x_{11} -x_{7} x_{10} ~\\2x_{7} x_{16} +2x_{1} x_{10} -6~\\2x_{7} x_{16} +2x_{3} x_{12} +2x_{1} x_{10} -8~\\2x_{7} x_{16} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -10~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -12~\\2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{1} x_{10} -14~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} +2x_{1} x_{10} -16~\\2x_{9} x_{18} +4x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} -18~\\x_{9} x_{18} +2x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} -9~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: C^{1}_3+3A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+4A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1+A^{1}_1 Containing regular semisimple subalgebra number 4: B^{1}_2+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 5: B^{1}_2+2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+4V_{4\psi}+4V_{3\psi}+15V_{2\psi}+16V_{\psi}+16V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{62}+1/10g_{58}+1/17g_{52}+1203616/2110347g_{44}+111022/2110347g_{39}-2845/4220694g_{34}+1518/1663g_{29}+267/3326g_{23}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{16} +x_{2} x_{15} -x_{1} x_{14} ~\\x_{8} x_{15} +x_{7} x_{10} -x_{6} x_{9} ~\\2x_{6} x_{14} +2x_{1} x_{9} -6~\\2x_{6} x_{14} +2x_{3} x_{11} +2x_{1} x_{9} -8~\\2x_{6} x_{14} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{1} x_{9} -10~\\2x_{6} x_{14} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{1} x_{9} -12~\\2x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -14~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} -16~\\2x_{8} x_{16} +4x_{7} x_{15} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} -16~\\x_{8} x_{16} +2x_{7} x_{15} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} -8~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1 Containing regular semisimple subalgebra number 4: B^{1}_2+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+3V_{4\psi}+6V_{3\psi}+10V_{2\psi}+18V_{\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 7h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{62}+1/10g_{58}+466471/902398g_{52}+34838/451199g_{48}-596/451199g_{44}+1437/1499g_{23}+180/1499g_{16}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{13} -x_{1} x_{12} ~\\x_{7} x_{13} +x_{6} x_{9} -x_{5} x_{8} ~\\2x_{5} x_{12} +2x_{1} x_{8} -6~\\2x_{5} x_{12} +2x_{3} x_{10} +2x_{1} x_{8} -8~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{1} x_{8} -10~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -12~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} -14~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} -14~\\2x_{7} x_{14} +4x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} -14~\\x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} -7~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: A^{2}_2+5A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_2+2A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+10V_{3\psi}+16V_{2\psi}+10V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 1/5g_{58}+1/10g_{52}+2g_{45}+1/17g_{44}+g_{36}+1/26g_{34}+1/37g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{8} -4~\\2x_{2} x_{9} +2x_{1} x_{8} -8~\\2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -10~\\2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -12~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -14~\\2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -16~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -18~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{1} x_{8} -9~\\\end{array}\)

h-characteristic: (2, 0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 3: B^{1}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{4\psi}+8V_{3\psi}+6V_{2\psi}+16V_{\psi}+37V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{62}+28192/67071g_{58}+8366/67071g_{55}-421/134142g_{52}+294/283g_{16}+111/566g_{9}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{12} +x_{2} x_{11} -x_{1} x_{10} ~\\x_{6} x_{11} +x_{5} x_{8} -x_{4} x_{7} ~\\2x_{4} x_{10} +2x_{1} x_{7} -6~\\2x_{4} x_{10} +2x_{3} x_{9} +2x_{1} x_{7} -8~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -10~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -12~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -12~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -12~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -12~\\x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} +x_{2} x_{8} -6~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: A^{2}_2+4A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_2+2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+8V_{3\psi}+15V_{2\psi}+16V_{\psi}+12V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 1/5g_{58}+1/10g_{52}+2g_{49}+1/17g_{44}+1/26g_{34}+g_{30}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -4~\\2x_{2} x_{8} +2x_{1} x_{7} -8~\\2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -10~\\2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -12~\\2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -14~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -16~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +4x_{1} x_{7} -16~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +2x_{1} x_{7} -8~\\\end{array}\)

h-characteristic: (2, 0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_3 Containing regular semisimple subalgebra number 2: B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+V_{4\psi}+10V_{3\psi}+3V_{2\psi}+10V_{\psi}+55V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 2111/11022g_{62}+1330/5511g_{60}-68/5511g_{58}+201/167g_{9}+60/167g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{10} +x_{2} x_{9} -x_{1} x_{8} ~\\x_{5} x_{9} +x_{4} x_{7} -x_{3} x_{6} ~\\2x_{3} x_{8} +2x_{1} x_{6} -6~\\2x_{4} x_{9} +2x_{2} x_{7} +2x_{1} x_{6} -8~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\x_{5} x_{10} +2x_{4} x_{9} +x_{2} x_{7} -5~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_2+3A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+6V_{3\psi}+15V_{2\psi}+18V_{\psi}+16V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 7h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 1/5g_{58}+2g_{53}+1/10g_{52}+1/17g_{44}+g_{24}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -4~\\2x_{2} x_{7} +2x_{1} x_{6} -8~\\2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -10~\\2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -12~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -14~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +4x_{1} x_{6} -14~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +4x_{1} x_{6} -14~\\x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +2x_{1} x_{6} -7~\\\end{array}\)

h-characteristic: (2, 1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+12V_{3\psi}+V_{2\psi}+78V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\)
\( e = 4g_{62}+3/2g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{4} -6~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\x_{1} x_{3} -4~\\\end{array}\)

h-characteristic: (0, 1, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_2+2A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+4V_{3\psi}+16V_{2\psi}+16V_{\psi}+25V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 1/5g_{58}+2g_{56}+1/10g_{52}+g_{17}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -4~\\2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -10~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -12~\\2x_{4} x_{8} +2x_{3} x_{7} +4x_{1} x_{5} -12~\\2x_{4} x_{8} +2x_{3} x_{7} +4x_{1} x_{5} -12~\\2x_{4} x_{8} +2x_{3} x_{7} +4x_{1} x_{5} -12~\\x_{4} x_{8} +x_{3} x_{7} +2x_{1} x_{5} -6~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 18
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+2V_{3\psi}+18V_{2\psi}+10V_{\psi}+39V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{59}+1/5g_{58}+g_{10}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{4} -4~\\2x_{2} x_{5} +2x_{1} x_{4} -8~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -10~\\2x_{3} x_{6} +4x_{1} x_{4} -10~\\2x_{3} x_{6} +4x_{1} x_{4} -10~\\2x_{3} x_{6} +4x_{1} x_{4} -10~\\2x_{3} x_{6} +4x_{1} x_{4} -10~\\x_{3} x_{6} +2x_{1} x_{4} -5~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: 8A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+6A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1+4A^{1}_1 Containing regular semisimple subalgebra number 4: 3A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 5: 4A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(36V_{2\psi}+28V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+14h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}+1/5g_{58}+1/10g_{52}+1/17g_{44}+1/26g_{34}+1/37g_{22}+1/50g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{9} -2~\\2x_{2} x_{10} +2x_{1} x_{9} -4~\\2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -6~\\2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -8~\\2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -10~\\2x_{6} x_{14} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -12~\\2x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -14~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -8~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+21V_{2\psi}+58V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{61}+g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\2x_{1} x_{3} -4~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 14
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: 7A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+5A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 4: 3A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(28V_{2\psi}+14V_{\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 7h_{8}+14h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}+1/5g_{58}+1/10g_{52}+1/17g_{44}+1/26g_{34}+1/37g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{8} -2~\\2x_{2} x_{9} +2x_{1} x_{8} -4~\\2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -6~\\2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -8~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -10~\\2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -12~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -14~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -7~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 12
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: 6A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+4A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 4: 3A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(21V_{2\psi}+24V_{\psi}+25V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}+1/5g_{58}+1/10g_{52}+1/17g_{44}+1/26g_{34}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{7} -2~\\2x_{2} x_{8} +2x_{1} x_{7} -4~\\2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -6~\\2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -8~\\2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -10~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -12~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -12~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -6~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 5A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(15V_{2\psi}+30V_{\psi}+31V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{8}+10h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}+1/5g_{58}+1/10g_{52}+1/17g_{44}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -2~\\2x_{2} x_{7} +2x_{1} x_{6} -4~\\2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -6~\\2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -8~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -10~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -10~\\2x_{5} x_{10} +2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -10~\\x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -5~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 4A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(10V_{2\psi}+32V_{\psi}+42V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}+1/5g_{58}+1/10g_{52}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -2~\\2x_{2} x_{6} +2x_{1} x_{5} -4~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -6~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -8~\\x_{4} x_{8} +x_{3} x_{7} +x_{2} x_{6} +x_{1} x_{5} -4~\\\end{array}\)

h-characteristic: (0, 0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 3A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(6V_{2\psi}+30V_{\psi}+58V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 3h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}+1/5g_{58}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{4} -2~\\2x_{2} x_{5} +2x_{1} x_{4} -4~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\x_{3} x_{6} +x_{2} x_{5} +x_{1} x_{4} -3~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{2\psi}+24V_{\psi}+79V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 2h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\)
\( e = g_{64}+1/2g_{62}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{3} -2~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\x_{2} x_{4} +x_{1} x_{3} -2~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{2\psi}+14V_{\psi}+105V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\)
\( e = g_{64}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\x_{1} x_{2} -1~\\\end{array}\)