\(sl(2)\)-subalgebras of \(E^{1}_8\)

\(E^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

Number of sl(2) subalgebras: 69.

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
\(A^{1240}_1\)	(2, 2, 2, 2, 2, 2, 2, 2)	(92, 136, 182, 270, 220, 168, 114, 58)	\(V_{58\psi}+V_{46\psi}+V_{38\psi}+V_{34\psi}+V_{26\psi}+V_{22\psi}+V_{14\psi}+V_{2\psi}\)	0	\(\displaystyle 0\)	2480	1240	E^{1}_8;	E^{1}_8;
\(A^{760}_1\)	(2, 2, 2, 0, 2, 2, 2, 2)	(72, 106, 142, 210, 172, 132, 90, 46)	\(V_{46\psi}+V_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{2\psi}\)	0	\(\displaystyle 0\)	1520	760	E^{1}_8;	E^{1}_8;
\(A^{520}_1\)	(2, 2, 2, 0, 2, 0, 2, 2)	(60, 88, 118, 174, 142, 108, 74, 38)	\(V_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+2V_{22\psi}+V_{18\psi}+V_{16\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi} +V_{2\psi}\)	0	\(\displaystyle 0\)	1040	520	E^{1}_8;	E^{1}_8;
\(A^{400}_1\)	(2, 0, 0, 2, 0, 2, 2, 2)	(52, 76, 102, 152, 124, 96, 66, 34)	\(V_{34\psi}+V_{28\psi}+2V_{26\psi}+V_{22\psi}+2V_{18\psi}+V_{16\psi}+V_{14\psi}+2V_{10\psi}+V_{8\psi}+2V_{2\psi}\)	0	\(\displaystyle 0\)	800	400	E^{1}_8; E^{1}_7+A^{1}_1;	E^{1}_8; E^{1}_7+A^{1}_1;
\(A^{399}_1\)	(2, 1, 1, 0, 1, 2, 2, 2)	(52, 76, 102, 151, 124, 96, 66, 34)	\(V_{34\psi}+2V_{27\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+V_{2\psi} +3V_{0}\)	3	\(\displaystyle A^{1}_1\)	798	399	E^{1}_7;	E^{1}_7;
\(A^{280}_1\)	(2, 0, 0, 2, 0, 2, 0, 2)	(44, 64, 86, 128, 104, 80, 54, 28)	\(V_{28\psi}+V_{26\psi}+2V_{22\psi}+2V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi} +V_{2\psi}\)	0	\(\displaystyle 0\)	560	280	E^{1}_8; D^{1}_8;	E^{1}_8; D^{1}_8;
\(A^{232}_1\)	(2, 0, 0, 2, 0, 0, 2, 2)	(40, 58, 78, 116, 94, 72, 50, 26)	\(V_{26\psi}+2V_{22\psi}+V_{20\psi}+V_{18\psi}+2V_{16\psi}+2V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{6\psi} +V_{4\psi}+2V_{2\psi}\)	0	\(\displaystyle 0\)	464	232	E^{1}_8; E^{1}_7+A^{1}_1;	E^{1}_8; E^{1}_7+A^{1}_1;
\(A^{231}_1\)	(2, 1, 1, 0, 1, 0, 2, 2)	(40, 58, 78, 115, 94, 72, 50, 26)	\(V_{26\psi}+V_{22\psi}+2V_{21\psi}+V_{18\psi}+V_{16\psi}+2V_{15\psi}+V_{14\psi}+2V_{11\psi}+2V_{10\psi} +V_{6\psi}+2V_{5\psi}+V_{2\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	462	231	E^{1}_7;	E^{1}_7;
\(A^{184}_1\)	(2, 0, 0, 2, 0, 0, 2, 0)	(36, 52, 70, 104, 84, 64, 44, 22)	\(2V_{22\psi}+V_{20\psi}+V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{12\psi}+4V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi} +3V_{2\psi}\)	0	\(\displaystyle 0\)	368	184	E^{1}_8; D^{1}_8;	E^{1}_8; D^{1}_8;
\(A^{182}_1\)	(2, 1, 1, 0, 1, 1, 0, 1)	(36, 52, 70, 103, 84, 64, 43, 22)	\(V_{22\psi}+2V_{21\psi}+V_{18\psi}+2V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi} +V_{6\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}\)	3	not computed	364	182	D^{1}_7;	D^{1}_7;
\(A^{160}_1\)	(0, 0, 0, 2, 0, 0, 2, 2)	(32, 48, 64, 96, 78, 60, 42, 22)	\(V_{22\psi}+2V_{18\psi}+3V_{16\psi}+3V_{14\psi}+3V_{10\psi}+3V_{8\psi}+2V_{6\psi}+V_{4\psi}+4V_{2\psi}\)	0	\(\displaystyle 0\)	320	160	E^{1}_8; E^{1}_7+A^{1}_1; E^{1}_6+A^{1}_2;	E^{1}_8; E^{1}_7+A^{1}_1; E^{1}_6+A^{1}_2;
\(A^{159}_1\)	(0, 1, 1, 0, 1, 0, 2, 2)	(32, 48, 64, 95, 78, 60, 42, 22)	\(V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi} +2V_{7\psi}+V_{6\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	318	159	E^{1}_7;	E^{1}_7;
\(A^{157}_1\)	(1, 0, 0, 1, 0, 1, 2, 2)	(32, 47, 63, 94, 77, 60, 42, 22)	\(V_{22\psi}+2V_{17\psi}+3V_{16\psi}+2V_{15\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi} +2V_{2\psi}+4V_{\psi}+3V_{0}\)	3	not computed	314	157	E^{1}_6+A^{1}_1;	E^{1}_6+A^{1}_1;
\(A^{156}_1\)	(2, 0, 0, 0, 0, 2, 2, 2)	(32, 46, 62, 92, 76, 60, 42, 22)	\(V_{22\psi}+7V_{16\psi}+V_{14\psi}+V_{10\psi}+7V_{8\psi}+V_{2\psi}+14V_{0}\)	14	not computed	312	156	E^{1}_6;	E^{1}_6;
\(A^{120}_1\)	(0, 0, 0, 2, 0, 0, 2, 0)	(28, 42, 56, 84, 68, 52, 36, 18)	\(2V_{18\psi}+V_{16\psi}+3V_{14\psi}+3V_{12\psi}+3V_{10\psi}+3V_{8\psi}+5V_{6\psi}+V_{4\psi}+3V_{2\psi}\)	0	\(\displaystyle 0\)	240	120	E^{1}_8; D^{1}_8; A^{1}_8;	E^{1}_8; D^{1}_8; A^{1}_8;
\(A^{112}_1\)	(2, 0, 0, 0, 2, 0, 0, 2)	(28, 40, 54, 80, 66, 50, 34, 18)	\(V_{18\psi}+2V_{16\psi}+3V_{14\psi}+V_{12\psi}+6V_{10\psi}+3V_{8\psi}+3V_{6\psi}+2V_{4\psi}+4V_{2\psi} +V_{0}\)	1	\(\displaystyle 0\)	224	112	E^{1}_7+A^{1}_1; D^{1}_6+2A^{1}_1; D^{1}_7;	E^{1}_7+A^{1}_1; D^{1}_6+2A^{1}_1; D^{1}_7;
\(A^{111}_1\)	(2, 0, 0, 1, 0, 1, 0, 2)	(28, 40, 54, 80, 65, 50, 34, 18)	\(V_{18\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{11\psi}+3V_{10\psi}+4V_{9\psi}+V_{8\psi}+2V_{6\psi} +2V_{5\psi}+V_{4\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	222	111	E^{1}_7; D^{1}_6+A^{1}_1;	E^{1}_7; D^{1}_6+A^{1}_1;
\(A^{110}_1\)	(2, 1, 1, 0, 0, 0, 1, 2)	(28, 40, 54, 79, 64, 49, 34, 18)	\(V_{18\psi}+4V_{15\psi}+V_{14\psi}+6V_{10\psi}+4V_{9\psi}+V_{6\psi}+4V_{5\psi}+V_{2\psi}+10V_{0}\)	10	not computed	220	110	D^{1}_6;	D^{1}_6;
\(A^{88}_1\)	(0, 0, 0, 2, 0, 0, 0, 2)	(24, 36, 48, 72, 58, 44, 30, 16)	\(V_{16\psi}+3V_{14\psi}+2V_{12\psi}+6V_{10\psi}+3V_{8\psi}+5V_{6\psi}+4V_{4\psi}+4V_{2\psi}\)	0	\(\displaystyle 0\)	176	88	E^{1}_8; D^{1}_8; E^{1}_6+A^{1}_2;	E^{1}_8; D^{1}_8; E^{1}_6+A^{1}_2;
\(A^{85}_1\)	(1, 0, 0, 1, 0, 1, 0, 2)	(24, 35, 47, 70, 57, 44, 30, 16)	\(V_{16\psi}+V_{14\psi}+2V_{13\psi}+2V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi} +V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}\)	1	\(\displaystyle 0\)	170	85	A^{1}_7+A^{1}_1; E^{1}_6+A^{1}_1;	A^{1}_7+A^{1}_1; E^{1}_6+A^{1}_1;
\(A^{84}_1\)	(2, 0, 0, 0, 0, 2, 0, 2)	(24, 34, 46, 68, 56, 44, 30, 16)	\(V_{16\psi}+V_{14\psi}+6V_{12\psi}+2V_{10\psi}+7V_{8\psi}+V_{6\psi}+7V_{4\psi}+V_{2\psi}+8V_{0}\)	8	\(\displaystyle A^{1}_2\)	168	84	A^{1}_7; E^{1}_6;	A^{1}_7; E^{1}_6;
\(A^{84}_1\)	(1, 0, 0, 1, 0, 1, 1, 0)	(24, 35, 47, 70, 57, 44, 30, 15)	\(2V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+V_{6\psi}+2V_{5\psi} +3V_{4\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}\)	3	not computed	168	84	A^{1}_7;	A^{1}_7;
\(A^{70}_1\)	(1, 0, 0, 1, 0, 1, 0, 1)	(22, 32, 43, 64, 52, 40, 27, 14)	\(V_{14\psi}+2V_{13\psi}+V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+3V_{6\psi} +2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}\)	1	\(\displaystyle 0\)	140	70	D^{1}_5+A^{1}_3; D^{1}_7;	D^{1}_5+A^{1}_3; D^{1}_7;
\(A^{64}_1\)	(0, 0, 0, 0, 2, 0, 0, 2)	(20, 30, 40, 60, 50, 38, 26, 14)	\(V_{14\psi}+2V_{12\psi}+7V_{10\psi}+5V_{8\psi}+5V_{6\psi}+5V_{4\psi}+8V_{2\psi}+V_{0}\)	1	\(\displaystyle 0\)	128	64	E^{1}_7+A^{1}_1; D^{1}_6+2A^{1}_1; D^{1}_5+A^{1}_2;	E^{1}_7+A^{1}_1; D^{1}_6+2A^{1}_1; D^{1}_5+A^{1}_2;
\(A^{63}_1\)	(0, 0, 0, 1, 0, 1, 0, 2)	(20, 30, 40, 60, 49, 38, 26, 14)	\(V_{14\psi}+V_{12\psi}+2V_{11\psi}+4V_{10\psi}+4V_{9\psi}+2V_{8\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+2V_{4\psi} +4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	126	63	E^{1}_7; D^{1}_6+A^{1}_1;	E^{1}_7; D^{1}_6+A^{1}_1;
\(A^{62}_1\)	(0, 1, 1, 0, 0, 0, 1, 2)	(20, 30, 40, 59, 48, 37, 26, 14)	\(V_{14\psi}+4V_{11\psi}+2V_{10\psi}+4V_{9\psi}+5V_{8\psi}+2V_{6\psi}+4V_{5\psi}+4V_{3\psi}+6V_{2\psi}+6V_{0}\)	6	\(\displaystyle 2A^{1}_1\)	124	62	D^{1}_5+2A^{1}_1; D^{1}_6;	D^{1}_5+2A^{1}_1; D^{1}_6;
\(A^{61}_1\)	(1, 0, 0, 0, 1, 0, 1, 2)	(20, 29, 39, 58, 48, 37, 26, 14)	\(V_{14\psi}+2V_{11\psi}+5V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+4V_{4\psi}+2V_{3\psi} +2V_{2\psi}+6V_{\psi}+6V_{0}\)	6	not computed	122	61	D^{1}_5+A^{1}_1;	D^{1}_5+A^{1}_1;
\(A^{60}_1\)	(2, 0, 0, 0, 0, 0, 2, 2)	(20, 28, 38, 56, 46, 36, 26, 14)	\(V_{14\psi}+9V_{10\psi}+7V_{8\psi}+V_{6\psi}+8V_{4\psi}+V_{2\psi}+21V_{0}\)	21	not computed	120	60	D^{1}_5;	D^{1}_5;
\(A^{57}_1\)	(1, 0, 0, 1, 0, 1, 0, 0)	(20, 29, 39, 58, 47, 36, 24, 12)	\(3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+4V_{5\psi}+3V_{4\psi}+2V_{3\psi} +2V_{2\psi}+2V_{\psi}+3V_{0}\)	3	not computed	114	57	A^{1}_6+A^{1}_1;	A^{1}_6+A^{1}_1;
\(A^{56}_1\)	(2, 0, 0, 0, 0, 2, 0, 0)	(20, 28, 38, 56, 46, 36, 24, 12)	\(3V_{12\psi}+5V_{10\psi}+3V_{8\psi}+13V_{6\psi}+3V_{4\psi}+5V_{2\psi}+6V_{0}\)	6	not computed	112	56	2D^{1}_4; A^{1}_6;	2D^{1}_4; A^{1}_6;
\(A^{40}_1\)	(0, 0, 0, 0, 2, 0, 0, 0)	(16, 24, 32, 48, 40, 30, 20, 10)	\(4V_{10\psi}+6V_{8\psi}+10V_{6\psi}+10V_{4\psi}+10V_{2\psi}\)	0	\(\displaystyle 0\)	80	40	E^{1}_8; D^{1}_8; E^{1}_7+A^{1}_1; E^{1}_6+A^{1}_2; D^{1}_6+2A^{1}_1; D^{1}_5+A^{1}_3; A^{1}_5+A^{1}_2+A^{1}_1; 2D^{1}_4; 2A^{1}_4;	E^{1}_8; D^{1}_8; E^{1}_7+A^{1}_1; E^{1}_6+A^{1}_2; D^{1}_6+2A^{1}_1; D^{1}_5+A^{1}_3; A^{1}_5+A^{1}_2+A^{1}_1; 2D^{1}_4; 2D^{1}_4; 2A^{1}_4;
\(A^{39}_1\)	(0, 0, 0, 1, 0, 1, 0, 0)	(16, 24, 32, 48, 39, 30, 20, 10)	\(3V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+4V_{4\psi}+6V_{3\psi}+6V_{2\psi}+3V_{0}\)	3	\(\displaystyle A^{1}_1\)	78	39	E^{1}_7; D^{1}_6+A^{1}_1; A^{1}_5+A^{1}_2;	E^{1}_7; D^{1}_6+A^{1}_1; A^{1}_5+A^{1}_2;
\(A^{38}_1\)	(0, 1, 1, 0, 0, 0, 1, 0)	(16, 24, 32, 47, 38, 29, 20, 10)	\(2V_{10\psi}+4V_{9\psi}+V_{8\psi}+4V_{7\psi}+7V_{6\psi}+4V_{5\psi}+5V_{4\psi}+8V_{3\psi}+3V_{2\psi}+6V_{0}\)	6	\(\displaystyle 2A^{1}_1\)	76	38	D^{1}_4+A^{1}_3; D^{1}_6;	D^{1}_4+A^{1}_3; D^{1}_6;
\(A^{37}_1\)	(1, 0, 0, 0, 1, 0, 1, 0)	(16, 23, 31, 46, 38, 29, 20, 10)	\(2V_{10\psi}+2V_{9\psi}+4V_{8\psi}+4V_{7\psi}+4V_{6\psi}+6V_{5\psi}+7V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi} +3V_{0}\)	3	not computed	74	37	E^{1}_6+A^{1}_1; A^{1}_5+2A^{1}_1;	E^{1}_6+A^{1}_1; A^{1}_5+2A^{1}_1;
\(A^{36}_1\)	(2, 0, 0, 0, 0, 0, 2, 0)	(16, 22, 30, 44, 36, 28, 20, 10)	\(2V_{10\psi}+8V_{8\psi}+8V_{6\psi}+15V_{4\psi}+3V_{2\psi}+14V_{0}\)	14	not computed	72	36	E^{1}_6; A^{1}_5+A^{1}_1;	E^{1}_6; A^{1}_5+A^{1}_1;
\(A^{36}_1\)	(1, 0, 0, 1, 0, 0, 0, 1)	(16, 23, 31, 46, 37, 28, 19, 10)	\(V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+5V_{6\psi}+8V_{5\psi}+7V_{4\psi}+4V_{3\psi}+2V_{2\psi}+4V_{\psi} +6V_{0}\)	6	not computed	72	36	A^{1}_5+A^{1}_1;	A^{1}_5+A^{1}_1;
\(A^{35}_1\)	(2, 0, 0, 0, 0, 1, 0, 1)	(16, 22, 30, 44, 36, 28, 19, 10)	\(V_{10\psi}+2V_{9\psi}+7V_{8\psi}+V_{6\psi}+14V_{5\psi}+7V_{4\psi}+2V_{3\psi}+V_{2\psi}+17V_{0}\)	17	not computed	70	35	A^{1}_5;	A^{1}_5;
\(A^{34}_1\)	(0, 0, 1, 0, 0, 1, 0, 1)	(15, 22, 30, 44, 36, 28, 19, 10)	\(V_{10\psi}+2V_{9\psi}+3V_{8\psi}+6V_{7\psi}+4V_{6\psi}+6V_{5\psi}+6V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi} +3V_{0}\)	3	not computed	68	34	D^{1}_5+A^{1}_2;	D^{1}_5+A^{1}_2;
\(A^{32}_1\)	(0, 2, 0, 0, 0, 0, 0, 2)	(14, 22, 28, 42, 34, 26, 18, 10)	\(V_{10\psi}+6V_{8\psi}+14V_{6\psi}+7V_{4\psi}+14V_{2\psi}+8V_{0}\)	8	not computed	64	32	D^{1}_4+4A^{1}_1; D^{1}_5+2A^{1}_1; D^{1}_4+A^{1}_2;	D^{1}_4+4A^{1}_1; D^{1}_5+2A^{1}_1; D^{1}_4+A^{1}_2;
\(A^{31}_1\)	(0, 0, 0, 1, 0, 0, 0, 2)	(14, 21, 28, 42, 34, 26, 18, 10)	\(V_{10\psi}+3V_{8\psi}+6V_{7\psi}+8V_{6\psi}+6V_{5\psi}+3V_{4\psi}+2V_{3\psi}+7V_{2\psi}+10V_{\psi}+6V_{0}\)	6	not computed	62	31	D^{1}_4+3A^{1}_1; D^{1}_5+A^{1}_1;	D^{1}_4+3A^{1}_1; D^{1}_5+A^{1}_1;
\(A^{30}_1\)	(1, 0, 0, 0, 0, 1, 0, 2)	(14, 20, 27, 40, 33, 26, 18, 10)	\(V_{10\psi}+V_{8\psi}+8V_{7\psi}+8V_{6\psi}+8V_{5\psi}+V_{4\psi}+8V_{2\psi}+8V_{\psi}+15V_{0}\)	15	\(\displaystyle A^{1}_3\)	60	30	D^{1}_4+2A^{1}_1; D^{1}_5;	D^{1}_4+2A^{1}_1; D^{1}_5;
\(A^{30}_1\)	(0, 0, 0, 1, 0, 0, 1, 0)	(14, 21, 28, 42, 34, 26, 18, 9)	\(2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+6V_{6\psi}+6V_{5\psi}+6V_{4\psi}+8V_{3\psi}+6V_{2\psi}+4V_{\psi}+3V_{0}\)	3	not computed	60	30	A^{1}_4+A^{1}_3;	A^{1}_4+A^{1}_3;
\(A^{29}_1\)	(0, 1, 0, 0, 0, 0, 1, 2)	(13, 20, 26, 39, 32, 25, 18, 10)	\(V_{10\psi}+6V_{7\psi}+14V_{6\psi}+6V_{5\psi}+2V_{2\psi}+14V_{\psi}+21V_{0}\)	21	not computed	58	29	D^{1}_4+A^{1}_1;	D^{1}_4+A^{1}_1;
\(A^{28}_1\)	(0, 0, 0, 0, 0, 0, 2, 2)	(12, 18, 24, 36, 30, 24, 18, 10)	\(V_{10\psi}+26V_{6\psi}+V_{2\psi}+52V_{0}\)	52	not computed	56	28	D^{1}_4;	D^{1}_4;
\(A^{25}_1\)	(0, 0, 1, 0, 0, 1, 0, 0)	(13, 19, 26, 38, 31, 24, 16, 8)	\(3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+10V_{4\psi}+8V_{3\psi}+7V_{2\psi}+6V_{\psi}+3V_{0}\)	3	not computed	50	25	A^{1}_4+A^{1}_2+A^{1}_1;	A^{1}_4+A^{1}_2+A^{1}_1;
\(A^{24}_1\)	(0, 0, 0, 0, 0, 2, 0, 0)	(12, 18, 24, 36, 30, 24, 16, 8)	\(3V_{8\psi}+13V_{6\psi}+14V_{4\psi}+18V_{2\psi}+6V_{0}\)	6	not computed	48	24	2D^{1}_4; A^{1}_4+A^{1}_2;	2D^{1}_4; A^{1}_4+A^{1}_2;
\(A^{22}_1\)	(0, 0, 0, 1, 0, 0, 0, 1)	(12, 18, 24, 36, 29, 22, 15, 8)	\(V_{8\psi}+4V_{7\psi}+5V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+9V_{2\psi}+8V_{\psi}+4V_{0}\)	4	not computed	44	22	2A^{1}_3+2A^{1}_1; D^{1}_4+A^{1}_3; A^{1}_4+2A^{1}_1;	2A^{1}_3+2A^{1}_1; D^{1}_4+A^{1}_3; A^{1}_4+2A^{1}_1;
\(A^{21}_1\)	(1, 0, 0, 0, 0, 1, 0, 1)	(12, 17, 23, 34, 28, 22, 15, 8)	\(V_{8\psi}+2V_{7\psi}+7V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+8V_{2\psi}+8V_{\psi}+9V_{0}\)	9	\(\displaystyle A^{1}_2\)	42	21	2A^{1}_3+A^{1}_1; A^{1}_4+A^{1}_1;	2A^{1}_3+A^{1}_1; A^{1}_4+A^{1}_1;
\(A^{20}_1\)	(2, 0, 0, 0, 0, 0, 0, 2)	(12, 16, 22, 32, 26, 20, 14, 8)	\(V_{8\psi}+11V_{6\psi}+21V_{4\psi}+11V_{2\psi}+24V_{0}\)	24	\(\displaystyle A^{1}_4\)	40	20	2A^{1}_3; A^{1}_4;	2A^{1}_3; A^{1}_4;
\(A^{20}_1\)	(1, 0, 0, 0, 1, 0, 0, 0)	(12, 17, 23, 34, 28, 21, 14, 7)	\(4V_{7\psi}+6V_{6\psi}+4V_{5\psi}+10V_{4\psi}+16V_{3\psi}+6V_{2\psi}+4V_{\psi}+10V_{0}\)	10	not computed	40	20	2A^{1}_3;	2A^{1}_3;
\(A^{16}_1\)	(0, 2, 0, 0, 0, 0, 0, 0)	(10, 16, 20, 30, 24, 18, 12, 6)	\(8V_{6\psi}+20V_{4\psi}+28V_{2\psi}+8V_{0}\)	8	not computed	32	16	D^{1}_4+4A^{1}_1; 4A^{1}_2; A^{1}_3+A^{1}_2+2A^{1}_1; D^{1}_4+A^{1}_2;	D^{1}_4+4A^{1}_1; 4A^{1}_2; A^{1}_3+A^{1}_2+2A^{1}_1; D^{1}_4+A^{1}_2;
\(A^{15}_1\)	(0, 0, 0, 1, 0, 0, 0, 0)	(10, 15, 20, 30, 24, 18, 12, 6)	\(5V_{6\psi}+6V_{5\psi}+10V_{4\psi}+14V_{3\psi}+15V_{2\psi}+10V_{\psi}+6V_{0}\)	6	not computed	30	15	D^{1}_4+3A^{1}_1; A^{1}_3+A^{1}_2+A^{1}_1;	D^{1}_4+3A^{1}_1; A^{1}_3+A^{1}_2+A^{1}_1;
\(A^{14}_1\)	(1, 0, 0, 0, 0, 1, 0, 0)	(10, 14, 19, 28, 23, 18, 12, 6)	\(3V_{6\psi}+8V_{5\psi}+8V_{4\psi}+16V_{3\psi}+16V_{2\psi}+8V_{\psi}+11V_{0}\)	11	not computed	28	14	A^{1}_3+4A^{1}_1; D^{1}_4+2A^{1}_1; A^{1}_3+A^{1}_2;	A^{1}_3+4A^{1}_1; D^{1}_4+2A^{1}_1; A^{1}_3+A^{1}_2;
\(A^{13}_1\)	(0, 1, 0, 0, 0, 0, 1, 0)	(9, 14, 18, 27, 22, 17, 12, 6)	\(2V_{6\psi}+6V_{5\psi}+13V_{4\psi}+12V_{3\psi}+16V_{2\psi}+14V_{\psi}+9V_{0}\)	9	\(\displaystyle 3A^{1}_1\)	26	13	3A^{1}_2+A^{1}_1; A^{1}_3+3A^{1}_1; D^{1}_4+A^{1}_1;	3A^{1}_2+A^{1}_1; A^{1}_3+3A^{1}_1; D^{1}_4+A^{1}_1;
\(A^{12}_1\)	(0, 0, 1, 0, 0, 0, 0, 1)	(9, 13, 18, 26, 21, 16, 11, 6)	\(V_{6\psi}+6V_{5\psi}+11V_{4\psi}+16V_{3\psi}+15V_{2\psi}+14V_{\psi}+13V_{0}\)	13	not computed	24	12	A^{1}_3+2A^{1}_1;	A^{1}_3+2A^{1}_1;
\(A^{12}_1\)	(0, 0, 0, 0, 0, 0, 2, 0)	(8, 12, 16, 24, 20, 16, 12, 6)	\(2V_{6\psi}+25V_{4\psi}+27V_{2\psi}+28V_{0}\)	28	\(\displaystyle D^{1}_4\)	24	12	3A^{1}_2; A^{1}_3+2A^{1}_1; D^{1}_4;	3A^{1}_2; A^{1}_3+2A^{1}_1; D^{1}_4;
\(A^{11}_1\)	(0, 0, 0, 0, 0, 1, 0, 1)	(8, 12, 16, 24, 20, 16, 11, 6)	\(V_{6\psi}+2V_{5\psi}+15V_{4\psi}+18V_{3\psi}+10V_{2\psi}+14V_{\psi}+24V_{0}\)	24	not computed	22	11	A^{1}_3+A^{1}_1;	A^{1}_3+A^{1}_1;
\(A^{10}_1\)	(1, 0, 0, 0, 0, 0, 0, 2)	(8, 11, 15, 22, 18, 14, 10, 6)	\(V_{6\psi}+11V_{4\psi}+32V_{3\psi}+V_{2\psi}+55V_{0}\)	55	not computed	20	10	A^{1}_3;	A^{1}_3;
\(A^{10}_1\)	(0, 0, 0, 0, 1, 0, 0, 0)	(8, 12, 16, 24, 20, 15, 10, 5)	\(4V_{5\psi}+10V_{4\psi}+16V_{3\psi}+20V_{2\psi}+20V_{\psi}+10V_{0}\)	10	not computed	20	10	2A^{1}_2+2A^{1}_1;	2A^{1}_2+2A^{1}_1;
\(A^{9}_1\)	(1, 0, 0, 0, 0, 0, 1, 0)	(8, 11, 15, 22, 18, 14, 10, 5)	\(2V_{5\psi}+10V_{4\psi}+16V_{3\psi}+23V_{2\psi}+18V_{\psi}+17V_{0}\)	17	not computed	18	9	2A^{1}_2+A^{1}_1;	2A^{1}_2+A^{1}_1;
\(A^{8}_1\)	(2, 0, 0, 0, 0, 0, 0, 0)	(8, 10, 14, 20, 16, 12, 8, 4)	\(14V_{4\psi}+50V_{2\psi}+28V_{0}\)	28	not computed	16	8	8A^{1}_1; A^{1}_2+4A^{1}_1; 2A^{1}_2;	8A^{1}_1; A^{1}_2+4A^{1}_1; 2A^{1}_2;
\(A^{7}_1\)	(0, 0, 1, 0, 0, 0, 0, 0)	(7, 10, 14, 20, 16, 12, 8, 4)	\(7V_{4\psi}+14V_{3\psi}+28V_{2\psi}+28V_{\psi}+17V_{0}\)	17	not computed	14	7	7A^{1}_1; A^{1}_2+3A^{1}_1;	7A^{1}_1; A^{1}_2+3A^{1}_1;
\(A^{6}_1\)	(0, 0, 0, 0, 0, 1, 0, 0)	(6, 9, 12, 18, 15, 12, 8, 4)	\(3V_{4\psi}+16V_{3\psi}+27V_{2\psi}+32V_{\psi}+24V_{0}\)	24	not computed	12	6	6A^{1}_1; A^{1}_2+2A^{1}_1;	6A^{1}_1; A^{1}_2+2A^{1}_1;
\(A^{5}_1\)	(1, 0, 0, 0, 0, 0, 0, 1)	(6, 8, 11, 16, 13, 10, 7, 4)	\(V_{4\psi}+12V_{3\psi}+32V_{2\psi}+32V_{\psi}+35V_{0}\)	35	\(\displaystyle A^{1}_5\)	10	5	5A^{1}_1; A^{1}_2+A^{1}_1;	5A^{1}_1; A^{1}_2+A^{1}_1;
\(A^{4}_1\)	(0, 0, 0, 0, 0, 0, 0, 2)	(4, 6, 8, 12, 10, 8, 6, 4)	\(V_{4\psi}+55V_{2\psi}+78V_{0}\)	78	\(\displaystyle E^{1}_6\)	8	4	4A^{1}_1; A^{1}_2;	4A^{1}_1; A^{1}_2;
\(A^{4}_1\)	(0, 1, 0, 0, 0, 0, 0, 0)	(5, 8, 10, 15, 12, 9, 6, 3)	\(8V_{3\psi}+28V_{2\psi}+48V_{\psi}+36V_{0}\)	36	not computed	8	4	4A^{1}_1;	4A^{1}_1;
\(A^{3}_1\)	(0, 0, 0, 0, 0, 0, 1, 0)	(4, 6, 8, 12, 10, 8, 6, 3)	\(2V_{3\psi}+27V_{2\psi}+52V_{\psi}+55V_{0}\)	55	not computed	6	3	3A^{1}_1;	3A^{1}_1;
\(A^{2}_1\)	(1, 0, 0, 0, 0, 0, 0, 0)	(4, 5, 7, 10, 8, 6, 4, 2)	\(14V_{2\psi}+64V_{\psi}+78V_{0}\)	78	not computed	4	2	2A^{1}_1;	2A^{1}_1;
\(A^{1}_1\)	(0, 0, 0, 0, 0, 0, 0, 1)	(2, 3, 4, 6, 5, 4, 3, 2)	\(V_{2\psi}+56V_{\psi}+133V_{0}\)	133	\(\displaystyle E^{1}_7\)	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 E^{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: 2480
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: E^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(V_{58\psi}+V_{46\psi}+V_{38\psi}+V_{34\psi}+V_{26\psi}+V_{22\psi}+V_{14\psi}+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 58h_{8}+114h_{7}+168h_{6}+220h_{5}+270h_{4}+182h_{3}+136h_{2}+92h_{1}\)
\( e = 29/25g_{8}+114/37g_{7}+84/13g_{6}+220/17g_{5}+27g_{4}+182/5g_{3}+68g_{2}+92g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{1} x_{9} -92~\\x_{2} x_{10} -136~\\x_{3} x_{11} -182~\\x_{4} x_{12} -270~\\x_{5} x_{13} -220~\\x_{6} x_{14} -168~\\x_{7} x_{15} -114~\\x_{8} x_{16} -58~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 2, 2, 2, 2)
Length of the weight dual to h: 1520
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: E^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(V_{46\psi}+V_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 46h_{8}+90h_{7}+132h_{6}+172h_{5}+210h_{4}+142h_{3}+106h_{2}+72h_{1}\)
\( e = 50579/4399g_{12}+37967/1590g_{11}-12137/996g_{10}+46/37g_{8}+45/13g_{7}+132/17g_{6}+3059/4399g_{5}+553/1590g_{3}+2597/996g_{2}+72g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{20} -x_{3} x_{19} -x_{2} x_{18} ~\\x_{10} x_{14} -x_{9} x_{13} -x_{8} x_{12} ~\\x_{1} x_{11} -72~\\x_{8} x_{18} +x_{2} x_{12} -106~\\x_{9} x_{19} +x_{3} x_{13} -142~\\x_{4} x_{14} +x_{3} x_{13} +x_{2} x_{12} -210~\\x_{10} x_{20} +x_{4} x_{14} -172~\\x_{5} x_{15} -132~\\x_{6} x_{16} -90~\\x_{7} x_{17} -46~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 2, 0, 2, 2)
Length of the weight dual to h: 1040
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: E^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(V_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+2V_{22\psi}+V_{18\psi}+V_{16\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}
+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 38h_{8}+74h_{7}+108h_{6}+142h_{5}+174h_{4}+118h_{3}+88h_{2}+60h_{1}\)
\( e = 1631055581639/1188940612995g_{20}+1260008858/578559909g_{14}+1047332629736/1188940612995g_{13}+223162354066/237788122599g_{12}+87438994/2777715g_{11}-33439087/902921g_{10}+19/13g_{8}+211814680/578559909g_{7}-11564221100/237788122599g_{5}-2188492/2777715g_{3}+3955006/902921g_{2}+60g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{10} x_{24} +x_{4} x_{21} -x_{3} x_{20} -x_{2} x_{19} ~\\-x_{9} x_{24} +x_{5} x_{23} -x_{4} x_{22} ~\\x_{12} x_{22} +x_{9} x_{16} -x_{8} x_{15} -x_{7} x_{14} ~\\-x_{12} x_{21} +x_{11} x_{17} -x_{10} x_{16} ~\\x_{1} x_{13} -60~\\x_{7} x_{19} +x_{2} x_{14} -88~\\x_{8} x_{20} +x_{3} x_{15} -118~\\x_{10} x_{22} +x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} -174~\\x_{12} x_{24} +x_{10} x_{22} +x_{9} x_{21} +x_{4} x_{16} -142~\\x_{9} x_{21} +x_{5} x_{17} +x_{4} x_{16} -108~\\x_{11} x_{23} +x_{5} x_{17} -74~\\x_{6} x_{18} -38~\\\end{array}\)

h-characteristic: (2, 0, 0, 2, 0, 2, 2, 2)
Length of the weight dual to h: 800
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: E^{1}_8 Containing regular semisimple subalgebra number 2: E^{1}_7+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{34\psi}+V_{28\psi}+2V_{26\psi}+V_{22\psi}+2V_{18\psi}+V_{16\psi}+V_{14\psi}+2V_{10\psi}+V_{8\psi}+2V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 34h_{8}+66h_{7}+96h_{6}+124h_{5}+152h_{4}+102h_{3}+76h_{2}+52h_{1}\)
\( e = -6850219020424825/1633719824936606g_{25}+420470631504504855/189511499692646296g_{19}+1688999805007181/1633719824936606g_{18}+1768546167279921/6534879299746424g_{17}+1320960/819797g_{13}-17740190481332631/189511499692646296g_{12}+3183933501905659/47377874923161574g_{11}+3403348864465371/32674396498732120g_{10}+4030780/1609721g_{9}+2g_{8}+33/5g_{7}+879216/819797g_{6}-1347852499030775/47377874923161574g_{4}+3064412/1609721g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{12} x_{28} +x_{9} x_{27} +x_{7} x_{25} +x_{2} x_{22} -x_{1} x_{20} ~\\x_{13} x_{28} +x_{9} x_{26} +x_{8} x_{25} +x_{2} x_{21} ~\\-x_{11} x_{28} -x_{8} x_{27} -x_{7} x_{26} +x_{3} x_{24} -x_{2} x_{23} ~\\x_{14} x_{26} +x_{13} x_{23} +x_{11} x_{21} +x_{8} x_{16} -x_{6} x_{15} ~\\x_{14} x_{27} +x_{12} x_{23} +x_{11} x_{22} +x_{7} x_{16} ~\\-x_{14} x_{25} -x_{13} x_{22} -x_{12} x_{21} +x_{10} x_{17} -x_{9} x_{16} ~\\x_{6} x_{20} +x_{1} x_{15} -52~\\x_{13} x_{27} +x_{9} x_{23} +x_{8} x_{22} +x_{2} x_{16} -76~\\x_{12} x_{26} +x_{9} x_{23} +x_{7} x_{21} +x_{2} x_{16} +x_{1} x_{15} -102~\\x_{14} x_{28} +x_{13} x_{27} +x_{12} x_{26} +x_{11} x_{25} +x_{9} x_{23} +x_{8} x_{22} +x_{7} x_{21} +x_{2} x_{16} -152~\\x_{11} x_{25} +x_{8} x_{22} +x_{7} x_{21} +x_{3} x_{17} +x_{2} x_{16} -124~\\x_{10} x_{24} +x_{3} x_{17} -96~\\x_{4} x_{18} -66~\\x_{5} x_{19} -34~\\\end{array}\)

h-characteristic: (2, 1, 1, 0, 1, 2, 2, 2)
Length of the weight dual to h: 798
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: E^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{34\psi}+2V_{27\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+V_{2\psi}
+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 34h_{8}+66h_{7}+96h_{6}+124h_{5}+151h_{4}+102h_{3}+76h_{2}+52h_{1}\)
\( e = 75/17g_{19}+49/2g_{18}+27/37g_{17}+34g_{8}+66/5g_{7}+48/5g_{6}+2g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{13} -52~\\x_{7} x_{14} +x_{2} x_{9} -76~\\x_{7} x_{14} +x_{5} x_{12} -102~\\x_{7} x_{14} +x_{5} x_{12} +x_{2} x_{9} -151~\\x_{5} x_{12} +x_{2} x_{9} -124~\\x_{4} x_{11} -96~\\x_{3} x_{10} -66~\\x_{1} x_{8} -34~\\\end{array}\)

h-characteristic: (2, 0, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 560
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: E^{1}_8 Containing regular semisimple subalgebra number 2: D^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(V_{28\psi}+V_{26\psi}+2V_{22\psi}+2V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi}
+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 28h_{8}+54h_{7}+80h_{6}+104h_{5}+128h_{4}+86h_{3}+64h_{2}+44h_{1}\)
\( e = -1066872149296115278282/162070670209921063259g_{25}-24862144996/64416575905g_{21}+976594854041262774742/486212010629763189777g_{19}+83103610030904244400/162070670209921063259g_{18}+64457080863532166896/162070670209921063259g_{17}+233481514/222381735g_{15}+135068663306/193249727715g_{14}+608258787526/1301214833281g_{13}+38135073553108202768/486212010629763189777g_{12}+211870191994543703536/1458636031889289569331g_{11}+36521056339453607162/162070670209921063259g_{10}+209414/34929g_{9}+778374688/4492111047g_{8}-36304704218/3903644499843g_{6}-121682775813939837250/1458636031889289569331g_{4}+78086/34929g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{13} x_{32} +x_{8} x_{30} +x_{6} x_{28} +x_{2} x_{23} -x_{1} x_{21} ~\\x_{14} x_{32} +x_{8} x_{29} +x_{7} x_{28} +x_{2} x_{22} ~\\-x_{12} x_{32} +x_{10} x_{31} -x_{7} x_{30} -x_{6} x_{29} +x_{3} x_{25} -x_{2} x_{24} ~\\-x_{9} x_{31} +x_{4} x_{27} -x_{3} x_{26} ~\\x_{16} x_{29} +x_{14} x_{24} +x_{12} x_{22} +x_{7} x_{18} -x_{5} x_{17} ~\\x_{16} x_{30} +x_{13} x_{24} +x_{12} x_{23} +x_{6} x_{18} ~\\-x_{16} x_{28} +x_{15} x_{26} -x_{14} x_{23} -x_{13} x_{22} +x_{9} x_{19} -x_{8} x_{18} ~\\-x_{15} x_{25} +x_{11} x_{20} -x_{10} x_{19} ~\\x_{5} x_{21} +x_{1} x_{17} -44~\\x_{14} x_{30} +x_{8} x_{24} +x_{7} x_{23} +x_{2} x_{18} -64~\\x_{13} x_{29} +x_{8} x_{24} +x_{6} x_{22} +x_{2} x_{18} +x_{1} x_{17} -86~\\x_{16} x_{32} +x_{14} x_{30} +x_{13} x_{29} +x_{12} x_{28} +x_{8} x_{24} +x_{7} x_{23} +x_{6} x_{22} +x_{2} x_{18} -128~\\x_{12} x_{28} +x_{10} x_{26} +x_{7} x_{23} +x_{6} x_{22} +x_{3} x_{19} +x_{2} x_{18} -104~\\x_{15} x_{31} +x_{10} x_{26} +x_{9} x_{25} +x_{3} x_{19} -80~\\x_{9} x_{25} +x_{4} x_{20} +x_{3} x_{19} -54~\\x_{11} x_{27} +x_{4} x_{20} -28~\\\end{array}\)

h-characteristic: (2, 0, 0, 2, 0, 0, 2, 2)
Length of the weight dual to h: 464
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: E^{1}_8 Containing regular semisimple subalgebra number 2: E^{1}_7+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{26\psi}+2V_{22\psi}+V_{20\psi}+V_{18\psi}+2V_{16\psi}+2V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{6\psi}
+V_{4\psi}+2V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 26h_{8}+50h_{7}+72h_{6}+94h_{5}+116h_{4}+78h_{3}+58h_{2}+40h_{1}\)
\( e = -6276857556469035454471/1401125927032108982529g_{33}+364037451058393289455/254750168551292542278g_{27}+4644461231836519938775/1401125927032108982529g_{26}+8202290907206394942641/3736335805418957286744g_{25}+238329415532446/282740957901639g_{21}-335550630538855893715/254750168551292542278g_{20}+97703444144009409565/301926125690420790848g_{19}-9464695122458289397505/3736335805418957286744g_{18}-14396508873038358710455/22418014832513743720464g_{17}+36178809194605/94246985967213g_{14}+981865544689510281289/905778377071262372544g_{12}-433553924555784747863/16304010787282722705792g_{11}+20648041233127228932967/22418014832513743720464g_{10}+4155690500/640829349g_{9}+13/5g_{8}+69300388523197/565481915803278g_{7}-6261181388660952450697/16304010787282722705792g_{4}+1263381380/640829349g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{16} x_{36} +x_{13} x_{35} +x_{11} x_{33} +x_{8} x_{30} +x_{6} x_{28} +x_{2} x_{25} -x_{1} x_{23} ~\\x_{17} x_{36} +x_{13} x_{34} +x_{12} x_{33} +x_{8} x_{29} +x_{7} x_{28} +x_{2} x_{24} ~\\-x_{10} x_{33} +x_{9} x_{32} -x_{7} x_{30} -x_{6} x_{29} -x_{2} x_{26} ~\\-x_{10} x_{36} -x_{7} x_{35} -x_{6} x_{34} +x_{3} x_{32} -x_{2} x_{31} ~\\-x_{15} x_{36} -x_{12} x_{35} -x_{11} x_{34} -x_{8} x_{31} +x_{3} x_{27} ~\\x_{18} x_{34} +x_{17} x_{31} +x_{15} x_{29} +x_{12} x_{26} +x_{10} x_{24} +x_{7} x_{20} -x_{5} x_{19} ~\\x_{18} x_{35} +x_{16} x_{31} +x_{15} x_{30} +x_{11} x_{26} +x_{10} x_{25} +x_{6} x_{20} ~\\-x_{15} x_{28} +x_{14} x_{27} -x_{12} x_{25} -x_{11} x_{24} -x_{8} x_{20} ~\\-x_{18} x_{33} -x_{17} x_{30} -x_{16} x_{29} -x_{13} x_{26} +x_{9} x_{21} ~\\-x_{18} x_{28} -x_{17} x_{25} -x_{16} x_{24} +x_{14} x_{21} -x_{13} x_{20} ~\\x_{5} x_{23} +x_{1} x_{19} -40~\\x_{17} x_{35} +x_{13} x_{31} +x_{12} x_{30} +x_{8} x_{26} +x_{7} x_{25} +x_{2} x_{20} -58~\\x_{16} x_{34} +x_{13} x_{31} +x_{11} x_{29} +x_{8} x_{26} +x_{6} x_{24} +x_{2} x_{20} +x_{1} x_{19} -78~\\x_{18} x_{36} +x_{17} x_{35} +x_{16} x_{34} +x_{15} x_{33} +x_{13} x_{31} +x_{12} x_{30} +x_{11} x_{29} +x_{10} x_{28} +x_{8} x_{26} +x_{7} x_{25} +x_{6} x_{24} +x_{2} x_{20} -116~\\x_{15} x_{33} +x_{12} x_{30} +x_{11} x_{29} +x_{10} x_{28} +x_{8} x_{26} +x_{7} x_{25} +x_{6} x_{24} +x_{3} x_{21} +x_{2} x_{20} -94~\\x_{10} x_{28} +x_{9} x_{27} +x_{7} x_{25} +x_{6} x_{24} +x_{3} x_{21} +x_{2} x_{20} -72~\\x_{14} x_{32} +x_{9} x_{27} +x_{3} x_{21} -50~\\x_{4} x_{22} -26~\\\end{array}\)

h-characteristic: (2, 1, 1, 0, 1, 0, 2, 2)
Length of the weight dual to h: 462
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: E^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{26\psi}+V_{22\psi}+2V_{21\psi}+V_{18\psi}+V_{16\psi}+2V_{15\psi}+V_{14\psi}+2V_{11\psi}+2V_{10\psi}
+V_{6\psi}+2V_{5\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 26h_{8}+50h_{7}+72h_{6}+94h_{5}+115h_{4}+78h_{3}+58h_{2}+40h_{1}\)
\( e = 7277/1320g_{27}-1517/456g_{26}+19/660g_{19}+269/228g_{18}+21/26g_{17}+2950/627g_{14}+26g_{8}+332/627g_{7}+40/17g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{4} x_{18} +x_{3} x_{17} -x_{2} x_{16} ~\\-x_{9} x_{13} +x_{8} x_{12} -x_{7} x_{11} ~\\x_{5} x_{14} -40~\\x_{7} x_{16} +x_{6} x_{15} +x_{2} x_{11} -58~\\x_{9} x_{18} +x_{6} x_{15} +x_{4} x_{13} -78~\\x_{9} x_{18} +x_{7} x_{16} +x_{6} x_{15} +x_{4} x_{13} +x_{2} x_{11} -115~\\x_{9} x_{18} +x_{7} x_{16} +x_{4} x_{13} +x_{2} x_{11} -94~\\x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} -72~\\x_{8} x_{17} +x_{3} x_{12} -50~\\x_{1} x_{10} -26~\\\end{array}\)

h-characteristic: (2, 0, 0, 2, 0, 0, 2, 0)
Length of the weight dual to h: 368
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: E^{1}_8 Containing regular semisimple subalgebra number 2: D^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{22\psi}+V_{20\psi}+V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{12\psi}+4V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi}
+3V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+44h_{7}+64h_{6}+84h_{5}+104h_{4}+70h_{3}+52h_{2}+36h_{1}\)
\( e = 38258440873429473783357285702/24337310727688890098200767283g_{33}-74695608720808932548718/81405326007456702560105g_{29}-21988085616804249336695986614/24337310727688890098200767283g_{27}-70835665323792194486754970551/48674621455377780196401534566g_{26}-122812706866742031437101028895/48674621455377780196401534566g_{25}+1414826348495461322122/16281065201491340512021g_{22}+5439925485129901824492/16281065201491340512021g_{21}+50945404083369112262988241467/48674621455377780196401534566g_{20}+114736439395415902771359527235/97349242910755560392803069132g_{19}+611165104785586732042548735015/778793943286044483142424553056g_{18}+41533573903317315799380412351/48674621455377780196401534566g_{17}+2130221916052031515924/16281065201491340512021g_{15}+557998036904193086956/16281065201491340512021g_{14}-311265937245827173613684211423/778793943286044483142424553056g_{12}-28047225614204503220642167763/97349242910755560392803069132g_{11}-105857000568117557397339362059/778793943286044483142424553056g_{10}+962712819/140565739g_{9}-324114878063877432178/16281065201491340512021g_{7}+57797204745582596388404838035/778793943286044483142424553056g_{4}+819530757/281131478g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{17} x_{40} +x_{13} x_{38} +x_{11} x_{36} +x_{7} x_{32} +x_{5} x_{30} +x_{2} x_{26} -x_{1} x_{24} ~\\x_{18} x_{40} +x_{13} x_{37} +x_{12} x_{36} +x_{7} x_{31} +x_{6} x_{30} +x_{2} x_{25} ~\\x_{15} x_{39} -x_{10} x_{36} +x_{8} x_{34} -x_{6} x_{32} -x_{5} x_{31} -x_{2} x_{27} ~\\-x_{10} x_{40} +x_{9} x_{39} -x_{6} x_{38} -x_{5} x_{37} +x_{3} x_{34} -x_{2} x_{33} ~\\-x_{16} x_{40} -x_{12} x_{38} -x_{11} x_{37} +x_{9} x_{35} -x_{7} x_{33} +x_{3} x_{28} ~\\-x_{14} x_{39} -x_{8} x_{35} -x_{3} x_{29} ~\\x_{20} x_{37} +x_{18} x_{33} +x_{16} x_{31} +x_{12} x_{27} +x_{10} x_{25} +x_{6} x_{22} -x_{4} x_{21} ~\\x_{20} x_{38} +x_{17} x_{33} +x_{16} x_{32} +x_{11} x_{27} +x_{10} x_{26} +x_{5} x_{22} ~\\x_{19} x_{35} -x_{16} x_{30} +x_{14} x_{28} -x_{12} x_{26} -x_{11} x_{25} -x_{7} x_{22} ~\\-x_{20} x_{36} -x_{18} x_{32} -x_{17} x_{31} +x_{15} x_{29} -x_{13} x_{27} +x_{8} x_{23} ~\\-x_{19} x_{34} -x_{15} x_{28} -x_{9} x_{23} ~\\-x_{20} x_{30} +x_{19} x_{29} -x_{18} x_{26} -x_{17} x_{25} +x_{14} x_{23} -x_{13} x_{22} ~\\x_{4} x_{24} +x_{1} x_{21} -36~\\x_{18} x_{38} +x_{13} x_{33} +x_{12} x_{32} +x_{7} x_{27} +x_{6} x_{26} +x_{2} x_{22} -52~\\x_{17} x_{37} +x_{13} x_{33} +x_{11} x_{31} +x_{7} x_{27} +x_{5} x_{25} +x_{2} x_{22} +x_{1} x_{21} -70~\\x_{20} x_{40} +x_{18} x_{38} +x_{17} x_{37} +x_{16} x_{36} +x_{13} x_{33} +x_{12} x_{32} +x_{11} x_{31} +x_{10} x_{30} +x_{7} x_{27} +x_{6} x_{26} +x_{5} x_{25} +x_{2} x_{22} -104~\\x_{16} x_{36} +x_{12} x_{32} +x_{11} x_{31} +x_{10} x_{30} +x_{9} x_{29} +x_{7} x_{27} +x_{6} x_{26} +x_{5} x_{25} +x_{3} x_{23} +x_{2} x_{22} -84~\\x_{15} x_{35} +x_{10} x_{30} +x_{9} x_{29} +x_{8} x_{28} +x_{6} x_{26} +x_{5} x_{25} +x_{3} x_{23} +x_{2} x_{22} -64~\\x_{19} x_{39} +x_{15} x_{35} +x_{14} x_{34} +x_{9} x_{29} +x_{8} x_{28} +x_{3} x_{23} -44~\\x_{14} x_{34} +x_{8} x_{28} +x_{3} x_{23} -22~\\\end{array}\)

h-characteristic: (2, 1, 1, 0, 1, 1, 0, 1)
Length of the weight dual to h: 364
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: D^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+2V_{21\psi}+V_{18\psi}+2V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}
+V_{6\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+43h_{7}+64h_{6}+84h_{5}+103h_{4}+70h_{3}+52h_{2}+36h_{1}\)
\( e = 21/26g_{28}+11g_{22}+6g_{19}+12g_{18}+40/17g_{17}+21/37g_{13}+18/5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{11} -36~\\x_{5} x_{12} +x_{1} x_{8} -52~\\x_{5} x_{12} +x_{3} x_{10} -70~\\x_{6} x_{13} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -103~\\x_{7} x_{14} +x_{6} x_{13} +x_{3} x_{10} +x_{1} x_{8} -84~\\x_{7} x_{14} +x_{6} x_{13} +x_{2} x_{9} -64~\\x_{6} x_{13} +x_{2} x_{9} -43~\\x_{2} x_{9} -22~\\\end{array}\)

h-characteristic: (0, 0, 0, 2, 0, 0, 2, 2)
Length of the weight dual to h: 320
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: E^{1}_8 Containing regular semisimple subalgebra number 2: E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 3: E^{1}_6+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+2V_{18\psi}+3V_{16\psi}+3V_{14\psi}+3V_{10\psi}+3V_{8\psi}+2V_{6\psi}+V_{4\psi}+4V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+42h_{7}+60h_{6}+78h_{5}+96h_{4}+64h_{3}+48h_{2}+32h_{1}\)
\( e = -2059624635409115819458603346233309180/939035662874879412181757010400902873g_{38}-16171363114616522825248553669204803328/939035662874879412181757010400902873g_{33}+1462305411481823118626613152819665747/939035662874879412181757010400902873g_{32}-617522638679793407860474842132074723/939035662874879412181757010400902873g_{30}+8009109413295598414842008213319528581/939035662874879412181757010400902873g_{27}+8376882629345726783936388169496639471/939035662874879412181757010400902873g_{26}-4023387204563639705636001028826127925/939035662874879412181757010400902873g_{25}+47170661937297842378468996349300976/313011887624959804060585670133634291g_{24}-341044563150751201631079135273804053/1878071325749758824363514020801805746g_{23}+5648059660654544/1645697630422947g_{21}-4858308846339889529466090706940044985/939035662874879412181757010400902873g_{20}+400681795391084351935498235571480424/313011887624959804060585670133634291g_{19}+2372137221514877775862815095728611457/939035662874879412181757010400902873g_{18}-1355958907098391179400208832916000363/1878071325749758824363514020801805746g_{17}+16461601351343796772966630172820495/313011887624959804060585670133634291g_{16}+651244891869698/1645697630422947g_{14}-187659419233584960872362833872082894/313011887624959804060585670133634291g_{12}+71184499583795620726927231485874933/313011887624959804060585670133634291g_{11}+1054731234710677779500191495669059133/1878071325749758824363514020801805746g_{10}+22/5g_{8}+77533609556956/548565876807649g_{7}-51418612336216728711568220093800652/313011887624959804060585670133634291g_{4}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{20} x_{44} +x_{18} x_{43} +x_{15} x_{41} +x_{12} x_{39} +x_{11} x_{38} +x_{8} x_{36} +x_{6} x_{32} +x_{5} x_{31} +x_{1} x_{26} ~\\x_{17} x_{43} +x_{12} x_{40} +x_{10} x_{38} +x_{6} x_{33} +x_{4} x_{31} +x_{1} x_{27} ~\\-x_{14} x_{41} -x_{9} x_{38} -x_{8} x_{37} +x_{7} x_{35} -x_{5} x_{33} -x_{4} x_{32} -x_{1} x_{28} ~\\x_{17} x_{44} +x_{12} x_{42} +x_{10} x_{41} +x_{6} x_{37} +x_{4} x_{36} +x_{1} x_{30} ~\\-x_{14} x_{44} -x_{9} x_{43} -x_{8} x_{42} -x_{5} x_{40} -x_{4} x_{39} +x_{2} x_{35} -x_{1} x_{34} ~\\-x_{19} x_{44} -x_{16} x_{43} -x_{15} x_{42} -x_{11} x_{40} -x_{10} x_{39} -x_{6} x_{34} +x_{2} x_{29} ~\\x_{22} x_{42} +x_{21} x_{40} +x_{19} x_{37} +x_{17} x_{34} +x_{16} x_{33} +x_{14} x_{30} +x_{10} x_{28} +x_{9} x_{27} +x_{4} x_{23} ~\\x_{21} x_{39} +x_{18} x_{34} +x_{16} x_{32} +x_{11} x_{28} +x_{9} x_{26} +x_{5} x_{23} ~\\x_{21} x_{44} +x_{18} x_{42} +x_{16} x_{41} +x_{11} x_{37} +x_{9} x_{36} +x_{5} x_{30} ~\\-x_{19} x_{36} -x_{16} x_{31} -x_{15} x_{30} +x_{13} x_{29} -x_{11} x_{27} -x_{10} x_{26} -x_{6} x_{23} ~\\-x_{22} x_{41} -x_{21} x_{38} -x_{20} x_{37} -x_{18} x_{33} -x_{17} x_{32} -x_{12} x_{28} +x_{7} x_{24} ~\\x_{22} x_{39} +x_{20} x_{34} +x_{19} x_{32} +x_{15} x_{28} +x_{14} x_{26} +x_{8} x_{23} ~\\x_{22} x_{43} +x_{20} x_{40} +x_{19} x_{38} +x_{15} x_{33} +x_{14} x_{31} +x_{8} x_{27} ~\\-x_{22} x_{36} -x_{21} x_{31} -x_{20} x_{30} -x_{18} x_{27} -x_{17} x_{26} +x_{13} x_{24} -x_{12} x_{23} ~\\x_{22} x_{44} +x_{20} x_{42} +x_{19} x_{41} +x_{15} x_{37} +x_{14} x_{36} +x_{8} x_{30} -32~\\x_{20} x_{42} +x_{18} x_{40} +x_{15} x_{37} +x_{12} x_{34} +x_{11} x_{33} +x_{8} x_{30} +x_{6} x_{28} +x_{5} x_{27} +x_{1} x_{23} -48~\\x_{22} x_{44} +x_{20} x_{42} +x_{19} x_{41} +x_{17} x_{39} +x_{15} x_{37} +x_{14} x_{36} +x_{12} x_{34} +x_{10} x_{32} +x_{8} x_{30} +x_{6} x_{28} +x_{4} x_{26} +x_{1} x_{23} -64~\\x_{22} x_{44} +x_{21} x_{43} +x_{20} x_{42} +x_{19} x_{41} +x_{18} x_{40} +x_{17} x_{39} +x_{16} x_{38} +x_{15} x_{37} +x_{14} x_{36} +x_{12} x_{34} +x_{11} x_{33} +x_{10} x_{32} +x_{9} x_{31} +x_{8} x_{30} +x_{6} x_{28} +x_{5} x_{27} +x_{4} x_{26} +x_{1} x_{23} -96~\\x_{19} x_{41} +x_{16} x_{38} +x_{15} x_{37} +x_{14} x_{36} +x_{11} x_{33} +x_{10} x_{32} +x_{9} x_{31} +x_{8} x_{30} +x_{6} x_{28} +x_{5} x_{27} +x_{4} x_{26} +x_{2} x_{24} +x_{1} x_{23} -78~\\x_{14} x_{36} +x_{9} x_{31} +x_{8} x_{30} +x_{7} x_{29} +x_{5} x_{27} +x_{4} x_{26} +x_{2} x_{24} +x_{1} x_{23} -60~\\x_{13} x_{35} +x_{7} x_{29} +x_{2} x_{24} -42~\\x_{3} x_{25} -22~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 1, 0, 2, 2)
Length of the weight dual to h: 318
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: E^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}
+2V_{7\psi}+V_{6\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+42h_{7}+60h_{6}+78h_{5}+95h_{4}+64h_{3}+48h_{2}+32h_{1}\)
\( e = 6960004229/233535680g_{32}-698740277/116767840g_{27}+3518183/620976g_{26}+217910099/46707136g_{24}-3062091/104257g_{23}-22044707/23353568g_{19}+469765/620976g_{18}+656438/104257g_{17}+975401/34935g_{14}+22g_{8}-18431/6987g_{7}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{9} x_{22} -x_{4} x_{19} +x_{3} x_{18} -x_{2} x_{17} ~\\x_{8} x_{22} +x_{5} x_{21} +x_{4} x_{20} ~\\-x_{11} x_{20} -x_{8} x_{15} +x_{7} x_{14} -x_{6} x_{13} ~\\x_{11} x_{19} +x_{10} x_{16} +x_{9} x_{15} ~\\x_{8} x_{19} +x_{5} x_{16} +x_{4} x_{15} -32~\\x_{10} x_{21} +x_{6} x_{17} +x_{5} x_{16} +x_{2} x_{13} -48~\\x_{11} x_{22} +x_{10} x_{21} +x_{9} x_{20} +x_{8} x_{19} +x_{5} x_{16} +x_{4} x_{15} -64~\\x_{11} x_{22} +x_{10} x_{21} +x_{9} x_{20} +x_{8} x_{19} +x_{6} x_{17} +x_{5} x_{16} +x_{4} x_{15} +x_{2} x_{13} -95~\\x_{11} x_{22} +x_{9} x_{20} +x_{8} x_{19} +x_{6} x_{17} +x_{4} x_{15} +x_{2} x_{13} -78~\\x_{9} x_{20} +x_{4} x_{15} +x_{3} x_{14} +x_{2} x_{13} -60~\\x_{7} x_{18} +x_{3} x_{14} -42~\\x_{1} x_{12} -22~\\\end{array}\)

h-characteristic: (1, 0, 0, 1, 0, 1, 2, 2)
Length of the weight dual to h: 314
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: E^{1}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+2V_{17\psi}+3V_{16\psi}+2V_{15\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}
+2V_{2\psi}+4V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+42h_{7}+60h_{6}+77h_{5}+94h_{4}+63h_{3}+47h_{2}+32h_{1}\)
\( e = 1/37g_{31}+16g_{30}+6g_{27}+30/17g_{26}+8/13g_{16}+11g_{8}+21/5g_{7}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{13} +x_{1} x_{8} -32~\\x_{7} x_{14} +x_{5} x_{12} +x_{1} x_{8} -47~\\x_{7} x_{14} +x_{6} x_{13} +x_{3} x_{10} +x_{1} x_{8} -63~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -94~\\x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -77~\\x_{5} x_{12} +x_{3} x_{10} -60~\\x_{4} x_{11} -42~\\x_{2} x_{9} -22~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 2, 2, 2)
Length of the weight dual to h: 312
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: E^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(V_{22\psi}+7V_{16\psi}+V_{14\psi}+V_{10\psi}+7V_{8\psi}+V_{2\psi}+14V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 22h_{8}+42h_{7}+60h_{6}+76h_{5}+92h_{4}+62h_{3}+46h_{2}+32h_{1}\)
\( e = 16g_{44}+30/17g_{27}+6g_{26}+11g_{8}+21/5g_{7}+8/13g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{12} +x_{1} x_{7} -32~\\x_{3} x_{9} +x_{1} x_{7} -46~\\x_{5} x_{11} +2x_{1} x_{7} -62~\\x_{5} x_{11} +x_{3} x_{9} +2x_{1} x_{7} -92~\\x_{5} x_{11} +x_{3} x_{9} +x_{1} x_{7} -76~\\x_{5} x_{11} +x_{3} x_{9} -60~\\x_{4} x_{10} -42~\\x_{2} x_{8} -22~\\\end{array}\)

h-characteristic: (0, 0, 0, 2, 0, 0, 2, 0)
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: 3
Containing regular semisimple subalgebra number 1: E^{1}_8 Containing regular semisimple subalgebra number 2: D^{1}_8 Containing regular semisimple subalgebra number 3: A^{1}_8
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{18\psi}+V_{16\psi}+3V_{14\psi}+3V_{12\psi}+3V_{10\psi}+3V_{8\psi}+5V_{6\psi}+V_{4\psi}+3V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+36h_{7}+52h_{6}+68h_{5}+84h_{4}+56h_{3}+42h_{2}+28h_{1}\)
\( e = 899836863609396109111320469238233590294515016/1078299247385151149702986835848990938741005371g_{38}+4165750292077950296757265815793190433418486628/1078299247385151149702986835848990938741005371g_{33}-95912772400684112671223801813924062377900116/1078299247385151149702986835848990938741005371g_{32}+179261070322480076685715175593375571923887548/1078299247385151149702986835848990938741005371g_{30}-11465461637680002431896/9924749845360598284031g_{29}-872801539459639966142024726118621689249230494/1078299247385151149702986835848990938741005371g_{27}-2896163858671965946703789583477317264469860448/1078299247385151149702986835848990938741005371g_{26}-502524245527877527252158110457126214819885906/1078299247385151149702986835848990938741005371g_{25}-81892578740025007850703605740175784478307984/1078299247385151149702986835848990938741005371g_{24}-43435060629098603017086845018549316952008902/1078299247385151149702986835848990938741005371g_{23}-751966303445121875070/9924749845360598284031g_{22}+3026235659042057983964/9924749845360598284031g_{21}+572469056177939451140614304351859708258340572/1078299247385151149702986835848990938741005371g_{20}+139964456336130200048823522264460929110808118/1078299247385151149702986835848990938741005371g_{19}-202849083223085681678401811578642675325233844/1078299247385151149702986835848990938741005371g_{18}-132764173441953230570924750536315031467989697/1078299247385151149702986835848990938741005371g_{17}+29470888470151528835965280405867010325018062/1078299247385151149702986835848990938741005371g_{16}+1525017547458096156746/9924749845360598284031g_{15}+778241068563917208147/9924749845360598284031g_{14}+153238460473499970169816991954950022022757516/1078299247385151149702986835848990938741005371g_{12}+150529973263220749705422302883060164143180531/1078299247385151149702986835848990938741005371g_{11}+150787098919856009683935076582973325109376502/1078299247385151149702986835848990938741005371g_{10}-181287855867213540550/9924749845360598284031g_{7}-94756299429223122261488013189219666432768678/1078299247385151149702986835848990938741005371g_{4}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{22} x_{48} +x_{19} x_{47} +x_{16} x_{45} +x_{12} x_{42} +x_{11} x_{41} +x_{8} x_{39} +x_{5} x_{34} +x_{4} x_{33} +x_{1} x_{27} ~\\x_{18} x_{47} +x_{12} x_{43} +x_{10} x_{41} +x_{5} x_{35} +x_{3} x_{33} +x_{1} x_{28} ~\\-x_{15} x_{45} +x_{14} x_{44} -x_{9} x_{41} -x_{8} x_{40} +x_{6} x_{37} -x_{4} x_{35} -x_{3} x_{34} -x_{1} x_{29} ~\\x_{18} x_{48} +x_{12} x_{46} +x_{10} x_{45} +x_{5} x_{40} +x_{3} x_{39} +x_{1} x_{32} ~\\-x_{15} x_{48} -x_{9} x_{47} -x_{8} x_{46} +x_{7} x_{44} -x_{4} x_{43} -x_{3} x_{42} +x_{2} x_{37} -x_{1} x_{36} ~\\-x_{21} x_{48} -x_{17} x_{47} -x_{16} x_{46} -x_{11} x_{43} -x_{10} x_{42} +x_{7} x_{38} -x_{5} x_{36} +x_{2} x_{30} ~\\-x_{13} x_{44} -x_{6} x_{38} -x_{2} x_{31} ~\\x_{24} x_{46} +x_{23} x_{43} +x_{21} x_{40} +x_{18} x_{36} +x_{17} x_{35} +x_{15} x_{32} +x_{10} x_{29} +x_{9} x_{28} +x_{3} x_{25} ~\\x_{23} x_{42} +x_{19} x_{36} +x_{17} x_{34} +x_{11} x_{29} +x_{9} x_{27} +x_{4} x_{25} ~\\x_{23} x_{48} +x_{19} x_{46} +x_{17} x_{45} +x_{11} x_{40} +x_{9} x_{39} +x_{4} x_{32} ~\\-x_{21} x_{39} +x_{20} x_{38} -x_{17} x_{33} -x_{16} x_{32} +x_{13} x_{30} -x_{11} x_{28} -x_{10} x_{27} -x_{5} x_{25} ~\\-x_{24} x_{45} -x_{23} x_{41} -x_{22} x_{40} -x_{19} x_{35} -x_{18} x_{34} +x_{14} x_{31} -x_{12} x_{29} +x_{6} x_{26} ~\\-x_{20} x_{37} -x_{14} x_{30} -x_{7} x_{26} ~\\x_{24} x_{42} +x_{22} x_{36} +x_{21} x_{34} +x_{16} x_{29} +x_{15} x_{27} +x_{8} x_{25} ~\\x_{24} x_{47} +x_{22} x_{43} +x_{21} x_{41} +x_{16} x_{35} +x_{15} x_{33} +x_{8} x_{28} ~\\-x_{24} x_{39} -x_{23} x_{33} -x_{22} x_{32} +x_{20} x_{31} -x_{19} x_{28} -x_{18} x_{27} +x_{13} x_{26} -x_{12} x_{25} ~\\x_{24} x_{48} +x_{22} x_{46} +x_{21} x_{45} +x_{16} x_{40} +x_{15} x_{39} +x_{8} x_{32} -28~\\x_{22} x_{46} +x_{19} x_{43} +x_{16} x_{40} +x_{12} x_{36} +x_{11} x_{35} +x_{8} x_{32} +x_{5} x_{29} +x_{4} x_{28} +x_{1} x_{25} -42~\\x_{24} x_{48} +x_{22} x_{46} +x_{21} x_{45} +x_{18} x_{42} +x_{16} x_{40} +x_{15} x_{39} +x_{12} x_{36} +x_{10} x_{34} +x_{8} x_{32} +x_{5} x_{29} +x_{3} x_{27} +x_{1} x_{25} -56~\\x_{24} x_{48} +x_{23} x_{47} +x_{22} x_{46} +x_{21} x_{45} +x_{19} x_{43} +x_{18} x_{42} +x_{17} x_{41} +x_{16} x_{40} +x_{15} x_{39} +x_{12} x_{36} +x_{11} x_{35} +x_{10} x_{34} +x_{9} x_{33} +x_{8} x_{32} +x_{5} x_{29} +x_{4} x_{28} +x_{3} x_{27} +x_{1} x_{25} -84~\\x_{21} x_{45} +x_{17} x_{41} +x_{16} x_{40} +x_{15} x_{39} +x_{11} x_{35} +x_{10} x_{34} +x_{9} x_{33} +x_{8} x_{32} +x_{7} x_{31} +x_{5} x_{29} +x_{4} x_{28} +x_{3} x_{27} +x_{2} x_{26} +x_{1} x_{25} -68~\\x_{15} x_{39} +x_{14} x_{38} +x_{9} x_{33} +x_{8} x_{32} +x_{7} x_{31} +x_{6} x_{30} +x_{4} x_{28} +x_{3} x_{27} +x_{2} x_{26} +x_{1} x_{25} -52~\\x_{20} x_{44} +x_{14} x_{38} +x_{13} x_{37} +x_{7} x_{31} +x_{6} x_{30} +x_{2} x_{26} -36~\\x_{13} x_{37} +x_{6} x_{30} +x_{2} x_{26} -18~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 2, 0, 0, 2)
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)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 3: D^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+2V_{16\psi}+3V_{14\psi}+V_{12\psi}+6V_{10\psi}+3V_{8\psi}+3V_{6\psi}+2V_{4\psi}+4V_{2\psi}
+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+34h_{7}+50h_{6}+66h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\)
\( e = -2179932399276223/1633719824936606g_{39}+1/17g_{35}+3/2g_{34}+634582057180277/6534879299746424g_{33}+481276620905319/1633719824936606g_{31}+1493389096209951/32674396498732120g_{25}+1395270/1609721g_{22}+130627216257608985/189511499692646296g_{20}+385280/819797g_{16}+1060758/1609721g_{15}+6192432920960145/189511499692646296g_{13}-1715269309002065/189511499692646296g_{12}+256438/819797g_{9}-3986200737431201/189511499692646296g_{5}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{12} x_{28} -x_{9} x_{27} -x_{7} x_{25} -x_{2} x_{22} +x_{1} x_{20} ~\\x_{13} x_{28} +x_{9} x_{26} +x_{8} x_{25} +x_{2} x_{21} ~\\x_{11} x_{28} +x_{8} x_{27} +x_{7} x_{26} -x_{3} x_{24} +x_{2} x_{23} ~\\-x_{14} x_{26} -x_{13} x_{23} -x_{11} x_{21} -x_{8} x_{16} +x_{6} x_{15} ~\\x_{14} x_{27} +x_{12} x_{23} +x_{11} x_{22} +x_{7} x_{16} ~\\x_{14} x_{25} +x_{13} x_{22} +x_{12} x_{21} -x_{10} x_{17} +x_{9} x_{16} ~\\x_{10} x_{24} +x_{3} x_{17} -28~\\x_{13} x_{27} +x_{9} x_{23} +x_{8} x_{22} +x_{4} x_{18} +x_{2} x_{16} -40~\\x_{13} x_{27} +x_{10} x_{24} +x_{9} x_{23} +x_{8} x_{22} +x_{5} x_{19} +x_{3} x_{17} +x_{2} x_{16} -54~\\x_{13} x_{27} +x_{11} x_{25} +x_{9} x_{23} +2x_{8} x_{22} +x_{7} x_{21} +x_{5} x_{19} +x_{4} x_{18} +x_{3} x_{17} +2x_{2} x_{16} -80~\\x_{14} x_{28} +x_{13} x_{27} +x_{12} x_{26} +x_{11} x_{25} +x_{9} x_{23} +x_{8} x_{22} +x_{7} x_{21} +x_{5} x_{19} +x_{4} x_{18} +x_{2} x_{16} -66~\\x_{12} x_{26} +x_{9} x_{23} +x_{7} x_{21} +x_{5} x_{19} +x_{4} x_{18} +x_{2} x_{16} +x_{1} x_{15} -50~\\x_{6} x_{20} +x_{5} x_{19} +x_{4} x_{18} +x_{1} x_{15} -34~\\x_{6} x_{20} +x_{1} x_{15} -18~\\\end{array}\)

h-characteristic: (2, 0, 0, 1, 0, 1, 0, 2)
Length of the weight dual to h: 222
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: E^{1}_7 Containing regular semisimple subalgebra number 2: D^{1}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{11\psi}+3V_{10\psi}+4V_{9\psi}+V_{8\psi}+2V_{6\psi}
+2V_{5\psi}+V_{4\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+34h_{7}+50h_{6}+65h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\)
\( e = -681661875562929885/320547995292679589g_{41}+334555652376772608/320547995292679589g_{35}+35053453758323060/320547995292679589g_{34}+134678147224441875/320547995292679589g_{33}+3/2g_{31}+26096257299267415/641095990585359178g_{28}-11359800176358714/320547995292679589g_{27}+21649668018215960/320547995292679589g_{26}-14773854601802593/641095990585359178g_{20}+579987/377126g_{15}+600474/863405g_{9}+365193/377126g_{8}+325738/863405g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{11} x_{26} -x_{8} x_{25} -x_{6} x_{23} -x_{2} x_{20} +x_{1} x_{18} ~\\x_{12} x_{26} +x_{8} x_{24} +x_{7} x_{23} +x_{2} x_{19} ~\\x_{10} x_{26} +x_{7} x_{25} +x_{6} x_{24} -x_{3} x_{22} +x_{2} x_{21} ~\\-x_{13} x_{24} -x_{12} x_{21} -x_{10} x_{19} -x_{7} x_{15} +x_{5} x_{14} ~\\x_{13} x_{25} +x_{11} x_{21} +x_{10} x_{20} +x_{6} x_{15} ~\\x_{13} x_{23} +x_{12} x_{20} +x_{11} x_{19} -x_{9} x_{16} +x_{8} x_{15} ~\\x_{9} x_{22} +x_{3} x_{16} -28~\\x_{12} x_{25} +x_{8} x_{21} +x_{7} x_{20} +x_{4} x_{17} +x_{2} x_{15} -40~\\x_{10} x_{23} +x_{7} x_{20} +x_{6} x_{19} +x_{4} x_{17} +x_{3} x_{16} +x_{2} x_{15} -54~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +2x_{4} x_{17} +x_{2} x_{15} -80~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +x_{4} x_{17} +x_{2} x_{15} -65~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +x_{2} x_{15} -50~\\x_{11} x_{24} +x_{8} x_{21} +x_{6} x_{19} +x_{2} x_{15} +x_{1} x_{14} -34~\\x_{5} x_{18} +x_{1} x_{14} -18~\\\end{array}\)

h-characteristic: (2, 1, 1, 0, 0, 0, 1, 2)
Length of the weight dual to h: 220
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: D^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+4V_{15\psi}+V_{14\psi}+6V_{10\psi}+4V_{9\psi}+V_{6\psi}+4V_{5\psi}+V_{2\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 18h_{8}+34h_{7}+49h_{6}+64h_{5}+79h_{4}+54h_{3}+40h_{2}+28h_{1}\)
\( e = 15/17g_{46}+24/5g_{35}+10g_{34}+15/26g_{17}+9g_{8}+14/5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{10} -28~\\x_{6} x_{12} +x_{5} x_{11} +x_{1} x_{7} -40~\\x_{6} x_{12} +x_{5} x_{11} +x_{3} x_{9} -54~\\x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} +x_{1} x_{7} -79~\\2x_{5} x_{11} +x_{3} x_{9} +x_{1} x_{7} -64~\\x_{5} x_{11} +x_{3} x_{9} +x_{1} x_{7} -49~\\x_{3} x_{9} +x_{1} x_{7} -34~\\x_{2} x_{8} -18~\\\end{array}\)

h-characteristic: (0, 0, 0, 2, 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: 3
Containing regular semisimple subalgebra number 1: E^{1}_8 Containing regular semisimple subalgebra number 2: D^{1}_8 Containing regular semisimple subalgebra number 3: E^{1}_6+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{16\psi}+3V_{14\psi}+2V_{12\psi}+6V_{10\psi}+3V_{8\psi}+5V_{6\psi}+4V_{4\psi}+4V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+30h_{7}+44h_{6}+58h_{5}+72h_{4}+48h_{3}+36h_{2}+24h_{1}\)
\( e = -1061555946374251982473441773215748344901290065/335297915197045302360177196482455026377095718g_{47}-1074200205868265107960072327816629872768336809/167648957598522651180088598241227513188547859g_{41}+8879772543253200871521384986402115218906515061/7041256219137951349563721126131555553919010078g_{40}-87426860740123192063133339679509868012614548/55882985866174217060029532747075837729515953g_{38}+8036243810553342103363106937516589985200203941/3520628109568975674781860563065777776959505039g_{35}+150450698641572552808968703476222656594447884/18627661955391405686676510915691945909838651g_{34}-372111480227253346362536598833490129908005532/55882985866174217060029532747075837729515953g_{33}+3561488374469810917752399706789107406712017658/3520628109568975674781860563065777776959505039g_{32}-3022138032007105427320077534551891116227343/37255323910782811373353021831383891819677302g_{30}+37574213173508119484440655/25779437491028674799853423g_{29}-1152494235442762436537242256948532758743100000/391180901063219519420206729229530864106611671g_{28}+12118607010811126292332038261389824986766174462/3520628109568975674781860563065777776959505039g_{27}+97971710077258350546163164575718292852742988/18627661955391405686676510915691945909838651g_{26}+10261671357828857170867060715997021827984410/18627661955391405686676510915691945909838651g_{25}-1044700047209326006631999657193753525024252737/7041256219137951349563721126131555553919010078g_{24}-40623408235895180525237184775731594409153499/335297915197045302360177196482455026377095718g_{23}+378903476629935899591501/1227592261477555942850163g_{22}-1217591985394615432978189669894114821440596732/391180901063219519420206729229530864106611671g_{20}-2201181590585097649101115954045831682510673098/3520628109568975674781860563065777776959505039g_{19}-741203243176345021099140106164277188390192/18627661955391405686676510915691945909838651g_{18}-52435004639111175366663393124703679386853980/167648957598522651180088598241227513188547859g_{17}+501767049279906613949266684295878494193283011/7041256219137951349563721126131555553919010078g_{16}+1047970137758539399412032/8593145830342891599951141g_{15}+208910217865236978557055895778554546211907212/391180901063219519420206729229530864106611671g_{12}+615680768444559732047331537745588591917099412/3520628109568975674781860563065777776959505039g_{11}+6337802364940882045456721447189450462838704/18627661955391405686676510915691945909838651g_{10}-778817831231162657024584/25779437491028674799853423g_{8}-77254120623601939069432665139865212361982484/391180901063219519420206729229530864106611671g_{4}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{26} x_{56} +x_{24} x_{55} +x_{21} x_{53} +x_{18} x_{51} +x_{17} x_{50} +x_{14} x_{48} +x_{11} x_{44} +x_{10} x_{43} +x_{7} x_{41} +x_{5} x_{37} +x_{4} x_{36} +x_{1} x_{31} ~\\x_{23} x_{55} +x_{18} x_{52} +x_{16} x_{50} +x_{11} x_{45} +x_{9} x_{43} +x_{5} x_{38} +x_{3} x_{36} +x_{1} x_{32} ~\\-x_{13} x_{48} +x_{12} x_{47} -x_{8} x_{43} -x_{7} x_{42} -x_{4} x_{38} -x_{3} x_{37} -x_{1} x_{33} ~\\x_{23} x_{56} +x_{18} x_{54} +x_{16} x_{53} +x_{11} x_{49} +x_{9} x_{48} +x_{5} x_{42} +x_{3} x_{41} +x_{1} x_{35} ~\\-x_{13} x_{53} -x_{8} x_{50} -x_{7} x_{49} +x_{6} x_{47} -x_{4} x_{45} -x_{3} x_{44} -x_{1} x_{39} ~\\-x_{13} x_{56} -x_{8} x_{55} -x_{7} x_{54} -x_{4} x_{52} -x_{3} x_{51} +x_{2} x_{47} -x_{1} x_{46} ~\\-x_{25} x_{56} -x_{22} x_{55} -x_{21} x_{54} -x_{17} x_{52} -x_{16} x_{51} -x_{11} x_{46} +x_{2} x_{34} ~\\-x_{20} x_{56} -x_{15} x_{55} -x_{14} x_{54} -x_{10} x_{52} -x_{9} x_{51} -x_{5} x_{46} +x_{2} x_{40} ~\\x_{28} x_{54} +x_{27} x_{52} +x_{25} x_{49} +x_{23} x_{46} +x_{22} x_{45} +x_{20} x_{42} +x_{16} x_{39} +x_{15} x_{38} +x_{13} x_{35} +x_{9} x_{33} +x_{8} x_{32} +x_{3} x_{29} ~\\x_{27} x_{51} +x_{24} x_{46} +x_{22} x_{44} +x_{17} x_{39} +x_{15} x_{37} +x_{10} x_{33} +x_{8} x_{31} +x_{4} x_{29} ~\\x_{27} x_{56} +x_{24} x_{54} +x_{22} x_{53} +x_{17} x_{49} +x_{15} x_{48} +x_{10} x_{42} +x_{8} x_{41} +x_{4} x_{35} ~\\-x_{20} x_{41} +x_{19} x_{40} -x_{15} x_{36} -x_{14} x_{35} -x_{10} x_{32} -x_{9} x_{31} -x_{5} x_{29} ~\\-x_{20} x_{53} -x_{15} x_{50} -x_{14} x_{49} -x_{10} x_{45} -x_{9} x_{44} +x_{6} x_{40} -x_{5} x_{39} ~\\-x_{28} x_{53} -x_{27} x_{50} -x_{26} x_{49} -x_{24} x_{45} -x_{23} x_{44} -x_{18} x_{39} +x_{6} x_{30} ~\\x_{28} x_{51} +x_{26} x_{46} +x_{25} x_{44} +x_{21} x_{39} +x_{20} x_{37} +x_{14} x_{33} +x_{13} x_{31} +x_{7} x_{29} ~\\x_{28} x_{55} +x_{26} x_{52} +x_{25} x_{50} +x_{21} x_{45} +x_{20} x_{43} +x_{14} x_{38} +x_{13} x_{36} +x_{7} x_{32} ~\\-x_{25} x_{41} -x_{22} x_{36} -x_{21} x_{35} +x_{19} x_{34} -x_{17} x_{32} -x_{16} x_{31} -x_{11} x_{29} ~\\-x_{25} x_{48} -x_{22} x_{43} -x_{21} x_{42} -x_{17} x_{38} -x_{16} x_{37} +x_{12} x_{34} -x_{11} x_{33} ~\\-x_{28} x_{48} -x_{27} x_{43} -x_{26} x_{42} -x_{24} x_{38} -x_{23} x_{37} -x_{18} x_{33} +x_{12} x_{30} ~\\-x_{28} x_{41} -x_{27} x_{36} -x_{26} x_{35} -x_{24} x_{32} -x_{23} x_{31} +x_{19} x_{30} -x_{18} x_{29} ~\\x_{28} x_{56} +x_{26} x_{54} +x_{25} x_{53} +x_{21} x_{49} +x_{20} x_{48} +x_{14} x_{42} +x_{13} x_{41} +x_{7} x_{35} -24~\\x_{26} x_{54} +x_{24} x_{52} +x_{21} x_{49} +x_{18} x_{46} +x_{17} x_{45} +x_{14} x_{42} +x_{11} x_{39} +x_{10} x_{38} +x_{7} x_{35} +x_{5} x_{33} +x_{4} x_{32} +x_{1} x_{29} -36~\\x_{28} x_{56} +x_{26} x_{54} +x_{25} x_{53} +x_{23} x_{51} +x_{21} x_{49} +x_{20} x_{48} +x_{18} x_{46} +x_{16} x_{44} +x_{14} x_{42} +x_{13} x_{41} +x_{11} x_{39} +x_{9} x_{37} +x_{7} x_{35} +x_{5} x_{33} +x_{3} x_{31} +x_{1} x_{29} -48~\\x_{28} x_{56} +x_{27} x_{55} +x_{26} x_{54} +x_{25} x_{53} +x_{24} x_{52} +x_{23} x_{51} +x_{22} x_{50} +x_{21} x_{49} +x_{20} x_{48} +x_{18} x_{46} +x_{17} x_{45} +x_{16} x_{44} +x_{15} x_{43} +x_{14} x_{42} +x_{13} x_{41} +x_{11} x_{39} +x_{10} x_{38} +x_{9} x_{37} +x_{8} x_{36} +x_{7} x_{35} +x_{5} x_{33} +x_{4} x_{32} +x_{3} x_{31} +x_{1} x_{29} -72~\\x_{25} x_{53} +x_{22} x_{50} +x_{21} x_{49} +x_{20} x_{48} +x_{17} x_{45} +x_{16} x_{44} +x_{15} x_{43} +x_{14} x_{42} +x_{13} x_{41} +x_{11} x_{39} +x_{10} x_{38} +x_{9} x_{37} +x_{8} x_{36} +x_{7} x_{35} +x_{5} x_{33} +x_{4} x_{32} +x_{3} x_{31} +x_{2} x_{30} +x_{1} x_{29} -58~\\x_{20} x_{48} +x_{15} x_{43} +x_{14} x_{42} +x_{13} x_{41} +x_{10} x_{38} +x_{9} x_{37} +x_{8} x_{36} +x_{7} x_{35} +x_{6} x_{34} +x_{5} x_{33} +x_{4} x_{32} +x_{3} x_{31} +x_{2} x_{30} +x_{1} x_{29} -44~\\x_{13} x_{41} +x_{12} x_{40} +x_{8} x_{36} +x_{7} x_{35} +x_{6} x_{34} +x_{4} x_{32} +x_{3} x_{31} +x_{2} x_{30} +x_{1} x_{29} -30~\\x_{19} x_{47} +x_{12} x_{40} +x_{6} x_{34} +x_{2} x_{30} -16~\\\end{array}\)

h-characteristic: (1, 0, 0, 1, 0, 1, 0, 2)
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: A^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: E^{1}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{16\psi}+V_{14\psi}+2V_{13\psi}+2V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}
+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\)
\( e = 7g_{41}+1/50g_{31}+7/37g_{28}+15/17g_{27}+3g_{26}+6g_{24}+6/13g_{23}+8/5g_{15}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{10} -24~\\x_{8} x_{16} +x_{6} x_{14} +x_{3} x_{11} +x_{1} x_{9} -35~\\x_{8} x_{16} +x_{6} x_{14} +x_{5} x_{13} +x_{2} x_{10} +x_{1} x_{9} -47~\\2x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -70~\\x_{8} x_{16} +x_{7} x_{15} +x_{5} x_{13} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -57~\\x_{7} x_{15} +x_{5} x_{13} +x_{3} x_{11} +x_{1} x_{9} -44~\\x_{7} x_{15} +x_{4} x_{12} +x_{1} x_{9} -30~\\x_{4} x_{12} -16~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 2, 0, 2)
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)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{1}_7 Containing regular semisimple subalgebra number 2: E^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(V_{16\psi}+V_{14\psi}+6V_{12\psi}+2V_{10\psi}+7V_{8\psi}+V_{6\psi}+7V_{4\psi}+V_{2\psi}+8V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{8}+30h_{7}+44h_{6}+56h_{5}+68h_{4}+46h_{3}+34h_{2}+24h_{1}\)
\( e = 7g_{54}+3g_{27}+15/17g_{26}+6/13g_{24}+6g_{23}+8/5g_{15}+7/37g_{14}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{13} +x_{2} x_{9} -24~\\x_{5} x_{12} +x_{2} x_{9} +x_{1} x_{8} -34~\\x_{6} x_{13} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -46~\\x_{6} x_{13} +x_{5} x_{12} +x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -68~\\x_{6} x_{13} +x_{5} x_{12} +x_{3} x_{10} +2x_{1} x_{8} -56~\\x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -44~\\x_{7} x_{14} +x_{4} x_{11} +x_{1} x_{8} -30~\\x_{4} x_{11} -16~\\\end{array}\)

h-characteristic: (1, 0, 0, 1, 0, 1, 1, 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: A^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+V_{6\psi}+2V_{5\psi}
+3V_{4\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 15h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\)
\( e = 7g_{33}+8/5g_{31}+6g_{24}+6/13g_{23}+3g_{22}+15/17g_{21}+7/37g_{20}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{13} +x_{2} x_{9} -24~\\x_{6} x_{13} +x_{4} x_{11} +x_{1} x_{8} -35~\\x_{6} x_{13} +x_{4} x_{11} +x_{2} x_{9} +x_{1} x_{8} -47~\\x_{7} x_{14} +x_{6} x_{13} +2x_{4} x_{11} +x_{2} x_{9} +x_{1} x_{8} -70~\\x_{7} x_{14} +x_{5} x_{12} +x_{4} x_{11} +x_{2} x_{9} +x_{1} x_{8} -57~\\x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -44~\\x_{5} x_{12} +x_{3} x_{10} -30~\\x_{3} x_{10} -15~\\\end{array}\)

h-characteristic: (1, 0, 0, 1, 0, 1, 0, 1)
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: D^{1}_5+A^{1}_3 Containing regular semisimple subalgebra number 2: D^{1}_7
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+2V_{13\psi}+V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+3V_{6\psi}
+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+27h_{7}+40h_{6}+52h_{5}+64h_{4}+43h_{3}+32h_{2}+22h_{1}\)
\( e = g_{35}+3/50g_{34}+3/26g_{33}+8g_{31}+7g_{29}+4/37g_{24}+18/5g_{23}+10/17g_{20}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{15} +x_{3} x_{11} -22~\\x_{8} x_{16} +x_{6} x_{14} +x_{3} x_{11} +x_{1} x_{9} -32~\\x_{7} x_{15} +x_{6} x_{14} +x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -43~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +2x_{1} x_{9} -64~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} +x_{1} x_{9} -52~\\x_{8} x_{16} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} -40~\\x_{8} x_{16} +x_{4} x_{12} +x_{2} x_{10} -27~\\x_{2} x_{10} -14~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 128
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: E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 3: D^{1}_5+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+2V_{12\psi}+7V_{10\psi}+5V_{8\psi}+5V_{6\psi}+5V_{4\psi}+8V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+38h_{6}+50h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\)
\( e = 38488272562541537767213/38198298141094823859369g_{51}+840182909768658/158305132736495g_{48}+1/10g_{47}-12609840446096746186003/9260193488750260329544g_{45}+30658906229611368017251/38198298141094823859369g_{44}+2988768024730975530419/9260193488750260329544g_{37}+124348677853769/158305132736495g_{35}-24251080846905390293645/13890290233125390494316g_{33}+28836336535121011095/203724256752505727249968g_{32}-62013071466099/158305132736495g_{28}+95623148847474242715471/74081547910002082636352g_{26}-7438619408884300672091/13890290233125390494316g_{25}+12665311226399788821705/203724256752505727249968g_{24}-1572447149/3337306477g_{22}-15644096460127887749103/74081547910002082636352g_{18}+2840866931/3337306477g_{15}+4947184236958357217249/148163095820004165272704g_{13}+6127167766576934654015/148163095820004165272704g_{5}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{16} x_{36} -x_{13} x_{35} -x_{11} x_{33} -x_{8} x_{30} -x_{6} x_{28} -x_{2} x_{25} +x_{1} x_{23} ~\\x_{17} x_{36} +x_{13} x_{34} -x_{12} x_{33} -x_{8} x_{29} -x_{7} x_{28} -x_{2} x_{24} ~\\x_{10} x_{33} +x_{9} x_{32} +x_{7} x_{30} +x_{6} x_{29} +x_{2} x_{26} ~\\x_{10} x_{36} -x_{7} x_{35} +x_{6} x_{34} +x_{3} x_{32} -x_{2} x_{31} ~\\x_{15} x_{36} -x_{12} x_{35} +x_{11} x_{34} -x_{8} x_{31} -x_{3} x_{27} ~\\-x_{18} x_{34} -x_{17} x_{31} -x_{15} x_{29} -x_{12} x_{26} -x_{10} x_{24} -x_{7} x_{20} +x_{5} x_{19} ~\\x_{18} x_{35} +x_{16} x_{31} -x_{15} x_{30} -x_{11} x_{26} -x_{10} x_{25} -x_{6} x_{20} ~\\x_{15} x_{28} +x_{14} x_{27} +x_{12} x_{25} +x_{11} x_{24} +x_{8} x_{20} ~\\x_{18} x_{33} -x_{17} x_{30} +x_{16} x_{29} -x_{13} x_{26} -x_{9} x_{21} ~\\x_{18} x_{28} -x_{17} x_{25} +x_{16} x_{24} +x_{14} x_{21} -x_{13} x_{20} ~\\x_{17} x_{35} +x_{13} x_{31} +x_{12} x_{30} +x_{8} x_{26} +x_{7} x_{25} +x_{4} x_{22} +x_{2} x_{20} -20~\\x_{15} x_{33} +x_{12} x_{30} +x_{11} x_{29} +x_{10} x_{28} +x_{8} x_{26} +x_{7} x_{25} +x_{6} x_{24} +x_{4} x_{22} +x_{3} x_{21} +x_{2} x_{20} -30~\\x_{17} x_{35} +x_{13} x_{31} +x_{12} x_{30} +x_{10} x_{28} +x_{9} x_{27} +x_{8} x_{26} +2x_{7} x_{25} +x_{6} x_{24} +x_{4} x_{22} +x_{3} x_{21} +2x_{2} x_{20} -40~\\x_{17} x_{35} +x_{15} x_{33} +x_{14} x_{32} +x_{13} x_{31} +2x_{12} x_{30} +x_{11} x_{29} +x_{10} x_{28} +x_{9} x_{27} +2x_{8} x_{26} +2x_{7} x_{25} +x_{6} x_{24} +x_{4} x_{22} +2x_{3} x_{21} +2x_{2} x_{20} -60~\\x_{18} x_{36} +x_{17} x_{35} +x_{16} x_{34} +x_{15} x_{33} +x_{14} x_{32} +x_{13} x_{31} +x_{12} x_{30} +x_{11} x_{29} +x_{10} x_{28} +x_{9} x_{27} +x_{8} x_{26} +x_{7} x_{25} +x_{6} x_{24} +x_{4} x_{22} +x_{3} x_{21} +x_{2} x_{20} -50~\\x_{16} x_{34} +x_{14} x_{32} +x_{13} x_{31} +x_{11} x_{29} +x_{9} x_{27} +x_{8} x_{26} +x_{6} x_{24} +x_{4} x_{22} +x_{3} x_{21} +x_{2} x_{20} +x_{1} x_{19} -38~\\x_{14} x_{32} +x_{9} x_{27} +x_{5} x_{23} +x_{4} x_{22} +x_{3} x_{21} +x_{1} x_{19} -26~\\x_{5} x_{23} +x_{1} x_{19} -14~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 1, 0, 2)
Length of the weight dual to h: 126
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: E^{1}_7 Containing regular semisimple subalgebra number 2: D^{1}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{12\psi}+2V_{11\psi}+4V_{10\psi}+4V_{9\psi}+2V_{8\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+2V_{4\psi}
+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+38h_{6}+49h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\)
\( e = 249384576069799/28923190369320g_{47}-98294082739/251876657825g_{44}-92342226508759/14461595184660g_{41}-1652072924287061/188000737400580g_{40}-1265094269611/106051698020840g_{38}-37832684717/503753315650g_{37}+282919688055754/47000184350145g_{35}+9220866033959/4820531728220g_{34}-4491792568817/53025849010420g_{33}+335464526190905/137867207427092g_{32}+29020212191/251876657825g_{31}-21326169442299/12533382493372g_{28}-49377628135402/34466801856773g_{27}+57647477019/964106345644g_{26}+4651257826203/12533382493372g_{20}+12102706/4585829g_{15}+5209890/4585829g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{15} x_{34} -x_{12} x_{33} -x_{10} x_{31} -x_{7} x_{28} -x_{5} x_{26} -x_{2} x_{23} +x_{1} x_{21} ~\\x_{16} x_{34} +x_{12} x_{32} +x_{11} x_{31} +x_{7} x_{27} +x_{6} x_{26} +x_{2} x_{22} ~\\x_{9} x_{31} +x_{8} x_{30} +x_{6} x_{28} +x_{5} x_{27} +x_{2} x_{24} ~\\x_{9} x_{34} +x_{6} x_{33} +x_{5} x_{32} -x_{3} x_{30} +x_{2} x_{29} ~\\x_{14} x_{34} +x_{11} x_{33} +x_{10} x_{32} +x_{7} x_{29} +x_{3} x_{25} ~\\-x_{17} x_{32} -x_{16} x_{29} -x_{14} x_{27} -x_{11} x_{24} -x_{9} x_{22} -x_{6} x_{19} +x_{4} x_{18} ~\\x_{17} x_{33} +x_{15} x_{29} +x_{14} x_{28} +x_{10} x_{24} +x_{9} x_{23} +x_{5} x_{19} ~\\x_{14} x_{26} +x_{13} x_{25} +x_{11} x_{23} +x_{10} x_{22} +x_{7} x_{19} ~\\x_{17} x_{31} +x_{16} x_{28} +x_{15} x_{27} +x_{12} x_{24} +x_{8} x_{20} ~\\x_{17} x_{26} +x_{16} x_{23} +x_{15} x_{22} -x_{13} x_{20} +x_{12} x_{19} ~\\x_{9} x_{26} +x_{8} x_{25} +x_{6} x_{23} +x_{5} x_{22} +x_{3} x_{20} +x_{2} x_{19} -20~\\x_{16} x_{33} +x_{13} x_{30} +x_{12} x_{29} +x_{11} x_{28} +x_{8} x_{25} +x_{7} x_{24} +x_{6} x_{23} +x_{3} x_{20} +x_{2} x_{19} -30~\\x_{14} x_{31} +x_{13} x_{30} +x_{11} x_{28} +x_{10} x_{27} +x_{9} x_{26} +x_{8} x_{25} +x_{7} x_{24} +x_{6} x_{23} +x_{5} x_{22} +2x_{3} x_{20} +x_{2} x_{19} -40~\\x_{17} x_{34} +x_{16} x_{33} +x_{15} x_{32} +x_{14} x_{31} +2x_{13} x_{30} +x_{12} x_{29} +x_{11} x_{28} +x_{10} x_{27} +x_{9} x_{26} +2x_{8} x_{25} +x_{7} x_{24} +x_{6} x_{23} +x_{5} x_{22} +2x_{3} x_{20} +x_{2} x_{19} -60~\\x_{17} x_{34} +x_{16} x_{33} +x_{15} x_{32} +x_{14} x_{31} +x_{13} x_{30} +x_{12} x_{29} +x_{11} x_{28} +x_{10} x_{27} +x_{9} x_{26} +x_{8} x_{25} +x_{7} x_{24} +x_{6} x_{23} +x_{5} x_{22} +x_{3} x_{20} +x_{2} x_{19} -49~\\x_{17} x_{34} +x_{16} x_{33} +x_{15} x_{32} +x_{14} x_{31} +x_{12} x_{29} +x_{11} x_{28} +x_{10} x_{27} +x_{9} x_{26} +x_{7} x_{24} +x_{6} x_{23} +x_{5} x_{22} +x_{2} x_{19} -38~\\x_{15} x_{32} +x_{12} x_{29} +x_{10} x_{27} +x_{7} x_{24} +x_{5} x_{22} +x_{2} x_{19} +x_{1} x_{18} -26~\\x_{4} x_{21} +x_{1} x_{18} -14~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 0, 0, 1, 2)
Length of the weight dual to h: 124
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: D^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+4V_{11\psi}+2V_{10\psi}+4V_{9\psi}+5V_{8\psi}+2V_{6\psi}+4V_{5\psi}+4V_{3\psi}+6V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+37h_{6}+48h_{5}+59h_{4}+40h_{3}+30h_{2}+20h_{1}\)
\( e = g_{46}+1/26g_{45}+18/5g_{40}+8g_{34}+1/37g_{30}+10/17g_{17}+7g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{6} x_{13} +x_{3} x_{10} -20~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{1} x_{8} -30~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} -40~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +2x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -59~\\x_{7} x_{14} +x_{6} x_{13} +2x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -48~\\x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -37~\\x_{3} x_{10} +x_{1} x_{8} -26~\\x_{2} x_{9} -14~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 1, 0, 1, 2)
Length of the weight dual to h: 122
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: D^{1}_5+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+2V_{11\psi}+5V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+4V_{4\psi}+2V_{3\psi}
+2V_{2\psi}+6V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+37h_{6}+48h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\)
\( e = g_{51}+8g_{48}+1/26g_{46}+10/17g_{37}+18/5g_{21}+7g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{11} +x_{4} x_{10} -20~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{1} x_{7} -29~\\x_{6} x_{12} +x_{5} x_{11} +2x_{4} x_{10} +x_{1} x_{7} -39~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{1} x_{7} -58~\\2x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -48~\\x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -37~\\x_{3} x_{9} +x_{1} x_{7} -26~\\x_{2} x_{8} -14~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 0, 2, 2)
Length of the weight dual to h: 120
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: D^{1}_5
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+9V_{10\psi}+7V_{8\psi}+V_{6\psi}+8V_{4\psi}+V_{2\psi}+21V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 14h_{8}+26h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\)
\( e = g_{69}+8g_{60}+10/17g_{32}+7g_{8}+18/5g_{7}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{10} +x_{4} x_{9} -20~\\2x_{4} x_{9} +x_{1} x_{6} -28~\\x_{5} x_{10} +2x_{4} x_{9} +x_{1} x_{6} -38~\\x_{5} x_{10} +3x_{4} x_{9} +2x_{1} x_{6} -56~\\x_{5} x_{10} +2x_{4} x_{9} +2x_{1} x_{6} -46~\\x_{5} x_{10} +x_{4} x_{9} +2x_{1} x_{6} -36~\\x_{3} x_{8} +x_{1} x_{6} -26~\\x_{2} x_{7} -14~\\\end{array}\)

h-characteristic: (1, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 114
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}_6+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+4V_{5\psi}+3V_{4\psi}+2V_{3\psi}
+2V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+36h_{6}+47h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\)
\( e = 6g_{50}+3/13g_{36}+12/5g_{34}+1/37g_{31}+6/5g_{27}+5g_{24}+10/17g_{23}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{12} +x_{2} x_{9} -20~\\x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -29~\\x_{7} x_{14} +x_{5} x_{12} +x_{4} x_{11} +x_{2} x_{9} +x_{1} x_{8} -39~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -58~\\x_{7} x_{14} +x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -47~\\x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -36~\\x_{6} x_{13} +x_{3} x_{10} +x_{1} x_{8} -24~\\x_{6} x_{13} +x_{1} x_{8} -12~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 2, 0, 0)
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: 2
Containing regular semisimple subalgebra number 1: 2D^{1}_4 Containing regular semisimple subalgebra number 2: A^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{12\psi}+5V_{10\psi}+3V_{8\psi}+13V_{6\psi}+3V_{4\psi}+5V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{8}+24h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\)
\( e = 6g_{62}+6/5g_{35}+6/17g_{34}+6/37g_{33}+5/13g_{24}+5g_{23}+3/25g_{22}+3/5g_{20}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{10} -20~\\x_{7} x_{15} +x_{5} x_{13} +x_{2} x_{10} +x_{1} x_{9} -28~\\x_{7} x_{15} +x_{6} x_{14} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -38~\\x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +2x_{1} x_{9} -56~\\x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +2x_{1} x_{9} -46~\\x_{8} x_{16} +x_{7} x_{15} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -36~\\x_{8} x_{16} +x_{5} x_{13} +x_{3} x_{11} +x_{1} x_{9} -24~\\x_{8} x_{16} +x_{1} x_{9} -12~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 2, 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)
Number of containing regular semisimple subalgebras: 10
Containing regular semisimple subalgebra number 1: E^{1}_8 Containing regular semisimple subalgebra number 2: D^{1}_8 Containing regular semisimple subalgebra number 3: E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 4: E^{1}_6+A^{1}_2 Containing regular semisimple subalgebra number 5: D^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 6: D^{1}_5+A^{1}_3 Containing regular semisimple subalgebra number 7: A^{1}_5+A^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 8: 2D^{1}_4 Containing regular semisimple subalgebra number 9: 2D^{1}_4 Containing regular semisimple subalgebra number 10: 2A^{1}_4
sl(2)-module decomposition of the ambient Lie algebra: \(4V_{10\psi}+6V_{8\psi}+10V_{6\psi}+10V_{4\psi}+10V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+30h_{6}+40h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\)
\( e = -1817483199688992058259714340394603652151745554267276407667151785130206143/7003451853884356753630106524905098697207387249044547205588639908006265414g_{66}+3245612922232513569495182954902734992249941813180242738777650782731231759/35017259269421783768150532624525493486036936245222736027943199540031327070g_{61}+14168412028290245844751548453593676981322779594338695951237519747009567729/56027614831074854029040852199240789577659097992356377644709119264050123312g_{58}+37458009121971505812031516709728943625644442598617796433929636784903292529/28013807415537427014520426099620394788829548996178188822354559632025061656g_{56}-170336019734968097636007979440211486354170463658903135152352828101630035/3501725926942178376815053262452549348603693624522273602794319954003132707g_{55}-18268281844645517323186486388631363753176017038371376201727544439001073643/140069037077687135072602130498101973944147744980890944111772798160125308280g_{53}-4224147509354737471058974076287527793205511276371808169861779131961065769/56027614831074854029040852199240789577659097992356377644709119264050123312g_{51}-5299994951171507094618358169446069467116906633852706026911263626911117211/7003451853884356753630106524905098697207387249044547205588639908006265414g_{50}+992322112836653302108246503692484688837511947308603587755209547288730293/35017259269421783768150532624525493486036936245222736027943199540031327070g_{49}-295146878127339962989404960738508771868044586827003397960441538583157131007/224110459324299416116163408796963158310636391969425510578836477056200493248g_{48}+921993032245336046834913308046739458525203667358370320512874191609864197/14006903707768713507260213049810197394414774498089094411177279816012530828g_{47}+9911195435849484544449861948582755214522757844436275517234607890737197303/140069037077687135072602130498101973944147744980890944111772798160125308280g_{45}+36612546494997984807581606899819333916425454654100524566003985227315177/3501725926942178376815053262452549348603693624522273602794319954003132707g_{44}+1888513697500137449268562172069413366358378261054083330432067013648808395/7003451853884356753630106524905098697207387249044547205588639908006265414g_{43}+2706681676270627988904574452972625003473290277799499755196677644576994677/14006903707768713507260213049810197394414774498089094411177279816012530828g_{42}+52255819329426645718121689123053029542365519812271700970564786446098860641/56027614831074854029040852199240789577659097992356377644709119264050123312g_{41}-4286944320398845036348329784623708078847956232105197971200512706424033687/280138074155374270145204260996203947888295489961781888223545596320250616560g_{40}+52574590445802639014044586167796535738409406580341177827260384534875461383/224110459324299416116163408796963158310636391969425510578836477056200493248g_{39}-532632512006033770484346993392334230231780289917184535228469841335586209/14006903707768713507260213049810197394414774498089094411177279816012530828g_{38}-62036619151988130424254479942967993815268675764801358299399365073702093/3501725926942178376815053262452549348603693624522273602794319954003132707g_{37}-15979265941210658024440054091866359678958839003324362912581924707024433381/140069037077687135072602130498101973944147744980890944111772798160125308280g_{36}-19142196696944377897110184558017998495102159734120797852967530979871198583/56027614831074854029040852199240789577659097992356377644709119264050123312g_{35}-15695415117686307276082611137852444797399076196443546309694342535120062419/56027614831074854029040852199240789577659097992356377644709119264050123312g_{34}-17534728545550542413517237054371534983553696339628880493782640842255312089/56027614831074854029040852199240789577659097992356377644709119264050123312g_{33}+481684645677119773200160684713133133783529886829510725121635626140473087/280138074155374270145204260996203947888295489961781888223545596320250616560g_{32}+772431179574968149597375312362757209106489721475634509609867074222927821/14006903707768713507260213049810197394414774498089094411177279816012530828g_{31}+37280307422105428033068154552762258076055360869921884044610189902108682/3501725926942178376815053262452549348603693624522273602794319954003132707g_{30}+220434775277035579718121927185917907380137803783201136808110981670685179/7003451853884356753630106524905098697207387249044547205588639908006265414g_{29}+183308924288298416988877144515460124686262251581849123507386082984815626319/1120552296621497080580817043984815791553181959847127552894182385281002466240g_{28}+7930675018065990242276653346771449594471096915861730994573087611627269743/56027614831074854029040852199240789577659097992356377644709119264050123312g_{27}+6516418421566593200438942779294403633412357851517406550867728190142180951/56027614831074854029040852199240789577659097992356377644709119264050123312g_{26}+97917433634605633908778357481755281456008959367317634754881488352066611/3501725926942178376815053262452549348603693624522273602794319954003132707g_{25}+963724277858187777218102089344069000089179457437436273633212364168265/3501725926942178376815053262452549348603693624522273602794319954003132707g_{24}-2049857597172457597204799799453777321991768260077263200904472653277729453/56027614831074854029040852199240789577659097992356377644709119264050123312g_{21}-121200127701544076995450697957336398185191875548030333641728984848622826359/1120552296621497080580817043984815791553181959847127552894182385281002466240g_{20}-82149381530470171828270521873248553857086238270976784042975788060721863/3501725926942178376815053262452549348603693624522273602794319954003132707g_{19}-44911106488180351658891499274875459186475663377434975392924629858291507/7003451853884356753630106524905098697207387249044547205588639908006265414g_{18}+2376563016839254489759614026477939174895374612510737435441557783731971813/56027614831074854029040852199240789577659097992356377644709119264050123312g_{13}+378647501928533301051036389394218423001016336818614759161161558946676187/14006903707768713507260213049810197394414774498089094411177279816012530828g_{12}-52469883265731838738082074150689779350624823098978109852341291624975407/3501725926942178376815053262452549348603693624522273602794319954003132707g_{5}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{36} x_{79} +x_{33} x_{77} +x_{30} x_{75} +x_{25} x_{72} +x_{22} x_{69} +x_{17} x_{64} +x_{14} x_{61} +x_{13} x_{60} +x_{9} x_{56} +x_{7} x_{52} +x_{3} x_{46} +x_{1} x_{42} ~\\-x_{28} x_{74} -x_{21} x_{69} -x_{16} x_{64} -x_{14} x_{62} -x_{9} x_{57} -x_{8} x_{55} -x_{5} x_{51} -x_{4} x_{50} -x_{2} x_{46} -x_{1} x_{43} ~\\x_{31} x_{79} +x_{27} x_{77} +x_{23} x_{75} +x_{19} x_{72} +x_{15} x_{69} -x_{13} x_{66} +x_{11} x_{64} +x_{8} x_{61} -x_{7} x_{58} +x_{5} x_{56} -x_{3} x_{50} -x_{1} x_{44} ~\\-x_{36} x_{80} -x_{30} x_{78} +x_{26} x_{77} -x_{22} x_{74} +x_{18} x_{72} -x_{14} x_{68} -x_{13} x_{67} +x_{10} x_{64} -x_{7} x_{59} +x_{4} x_{56} -x_{3} x_{51} -x_{1} x_{45} ~\\-x_{28} x_{78} -x_{21} x_{75} -x_{16} x_{72} -x_{14} x_{70} -x_{9} x_{65} -x_{8} x_{63} -x_{5} x_{59} -x_{4} x_{58} -x_{2} x_{52} -x_{1} x_{47} ~\\x_{33} x_{80} +x_{26} x_{79} +x_{25} x_{78} +x_{18} x_{75} +x_{17} x_{74} -x_{13} x_{71} +x_{10} x_{69} +x_{9} x_{68} -x_{7} x_{63} +x_{4} x_{61} -x_{3} x_{55} -x_{1} x_{48} ~\\-x_{31} x_{80} -x_{23} x_{78} +x_{20} x_{77} -x_{15} x_{74} +x_{13} x_{73} +x_{12} x_{72} -x_{8} x_{68} +x_{7} x_{65} +x_{6} x_{64} +x_{3} x_{57} +x_{2} x_{56} +x_{1} x_{49} ~\\-x_{28} x_{80} -x_{21} x_{79} -x_{16} x_{77} -x_{14} x_{76} -x_{9} x_{73} -x_{8} x_{71} -x_{5} x_{67} -x_{4} x_{66} -x_{2} x_{60} -x_{1} x_{53} ~\\x_{27} x_{80} +x_{20} x_{79} +x_{19} x_{78} +x_{13} x_{76} +x_{12} x_{75} +x_{11} x_{74} +x_{7} x_{70} +x_{6} x_{69} +x_{5} x_{68} +x_{3} x_{62} +x_{2} x_{61} +x_{1} x_{54} ~\\x_{39} x_{76} +x_{37} x_{73} +x_{35} x_{70} +x_{32} x_{65} +x_{29} x_{62} +x_{24} x_{57} +x_{21} x_{54} +x_{20} x_{53} +x_{16} x_{49} +x_{12} x_{47} +x_{6} x_{43} +x_{2} x_{41} ~\\x_{31} x_{76} +x_{27} x_{73} +x_{23} x_{70} +x_{20} x_{66} +x_{19} x_{65} +x_{15} x_{62} +x_{12} x_{58} +x_{11} x_{57} +x_{8} x_{54} +x_{6} x_{50} +x_{5} x_{49} +x_{2} x_{44} ~\\x_{39} x_{80} +x_{35} x_{78} +x_{29} x_{74} +x_{26} x_{73} +x_{21} x_{68} +x_{20} x_{67} +x_{18} x_{65} +x_{12} x_{59} +x_{10} x_{57} +x_{6} x_{51} +x_{4} x_{49} +x_{2} x_{45} ~\\x_{37} x_{80} +x_{32} x_{78} -x_{26} x_{76} +x_{24} x_{74} -x_{20} x_{71} -x_{18} x_{70} +x_{16} x_{68} -x_{12} x_{63} -x_{10} x_{62} -x_{6} x_{55} -x_{4} x_{54} -x_{2} x_{48} ~\\-x_{34} x_{68} -x_{29} x_{61} -x_{24} x_{56} -x_{22} x_{54} -x_{17} x_{49} -x_{15} x_{48} -x_{11} x_{45} -x_{10} x_{44} -x_{6} x_{42} -x_{3} x_{41} ~\\-x_{34} x_{78} -x_{29} x_{75} -x_{24} x_{72} -x_{22} x_{70} -x_{17} x_{65} -x_{15} x_{63} -x_{11} x_{59} -x_{10} x_{58} -x_{6} x_{52} -x_{3} x_{47} ~\\-x_{34} x_{80} -x_{29} x_{79} -x_{24} x_{77} -x_{22} x_{76} -x_{17} x_{73} -x_{15} x_{71} -x_{11} x_{67} -x_{10} x_{66} -x_{6} x_{60} -x_{3} x_{53} ~\\x_{39} x_{71} +x_{37} x_{67} +x_{35} x_{63} +x_{32} x_{59} +x_{29} x_{55} -x_{26} x_{53} +x_{24} x_{51} +x_{21} x_{48} -x_{18} x_{47} +x_{16} x_{45} -x_{10} x_{43} -x_{4} x_{41} ~\\x_{36} x_{71} +x_{33} x_{67} +x_{30} x_{63} +x_{26} x_{60} +x_{25} x_{59} +x_{22} x_{55} +x_{18} x_{52} +x_{17} x_{51} +x_{14} x_{48} +x_{10} x_{46} +x_{9} x_{45} +x_{4} x_{42} ~\\-x_{40} x_{76} -x_{38} x_{70} +x_{37} x_{66} -x_{34} x_{62} +x_{32} x_{58} -x_{28} x_{54} -x_{27} x_{53} +x_{24} x_{50} -x_{19} x_{47} +x_{16} x_{44} -x_{11} x_{43} -x_{5} x_{41} ~\\x_{40} x_{79} +x_{38} x_{75} +x_{34} x_{69} +x_{33} x_{66} +x_{28} x_{61} +x_{27} x_{60} +x_{25} x_{58} +x_{19} x_{52} +x_{17} x_{50} +x_{11} x_{46} +x_{9} x_{44} +x_{5} x_{42} ~\\x_{37} x_{79} +x_{33} x_{76} +x_{32} x_{75} +x_{27} x_{71} +x_{25} x_{70} +x_{24} x_{69} +x_{19} x_{63} +x_{17} x_{62} +x_{16} x_{61} +x_{11} x_{55} +x_{9} x_{54} +x_{5} x_{48} ~\\-x_{38} x_{68} -x_{35} x_{61} -x_{32} x_{56} -x_{30} x_{54} -x_{25} x_{49} -x_{23} x_{48} -x_{19} x_{45} -x_{18} x_{44} -x_{12} x_{42} -x_{7} x_{41} ~\\-x_{38} x_{74} -x_{35} x_{69} -x_{32} x_{64} -x_{30} x_{62} -x_{25} x_{57} -x_{23} x_{55} -x_{19} x_{51} -x_{18} x_{50} -x_{12} x_{46} -x_{7} x_{43} ~\\-x_{38} x_{80} -x_{35} x_{79} -x_{32} x_{77} -x_{30} x_{76} -x_{25} x_{73} -x_{23} x_{71} -x_{19} x_{67} -x_{18} x_{66} -x_{12} x_{60} -x_{7} x_{53} ~\\x_{40} x_{73} +x_{39} x_{66} +x_{38} x_{65} +x_{35} x_{58} +x_{34} x_{57} -x_{31} x_{53} +x_{29} x_{50} +x_{28} x_{49} -x_{23} x_{47} +x_{21} x_{44} -x_{15} x_{43} -x_{8} x_{41} ~\\x_{40} x_{77} +x_{38} x_{72} -x_{36} x_{66} +x_{34} x_{64} -x_{31} x_{60} -x_{30} x_{58} +x_{28} x_{56} -x_{23} x_{52} -x_{22} x_{50} -x_{15} x_{46} -x_{14} x_{44} -x_{8} x_{42} ~\\x_{39} x_{77} +x_{36} x_{73} +x_{35} x_{72} +x_{31} x_{67} +x_{30} x_{65} +x_{29} x_{64} +x_{23} x_{59} +x_{22} x_{57} +x_{21} x_{56} +x_{15} x_{51} +x_{14} x_{49} +x_{8} x_{45} ~\\-x_{40} x_{71} -x_{38} x_{63} +x_{37} x_{60} -x_{34} x_{55} +x_{33} x_{53} +x_{32} x_{52} -x_{28} x_{48} +x_{25} x_{47} +x_{24} x_{46} +x_{17} x_{43} +x_{16} x_{42} +x_{9} x_{41} ~\\-x_{40} x_{68} -x_{39} x_{61} -x_{37} x_{56} -x_{36} x_{54} -x_{33} x_{49} -x_{31} x_{48} -x_{27} x_{45} -x_{26} x_{44} -x_{20} x_{42} -x_{13} x_{41} ~\\-x_{40} x_{74} -x_{39} x_{69} -x_{37} x_{64} -x_{36} x_{62} -x_{33} x_{57} -x_{31} x_{55} -x_{27} x_{51} -x_{26} x_{50} -x_{20} x_{46} -x_{13} x_{43} ~\\-x_{40} x_{78} -x_{39} x_{75} -x_{37} x_{72} -x_{36} x_{70} -x_{33} x_{65} -x_{31} x_{63} -x_{27} x_{59} -x_{26} x_{58} -x_{20} x_{52} -x_{13} x_{47} ~\\x_{40} x_{67} +x_{39} x_{60} +x_{38} x_{59} +x_{36} x_{53} +x_{35} x_{52} +x_{34} x_{51} +x_{30} x_{47} +x_{29} x_{46} +x_{28} x_{45} +x_{22} x_{43} +x_{21} x_{42} +x_{14} x_{41} ~\\x_{40} x_{80} +x_{39} x_{79} +x_{38} x_{78} +x_{36} x_{76} +x_{35} x_{75} +x_{34} x_{74} +x_{31} x_{71} +x_{30} x_{70} +x_{29} x_{69} +x_{28} x_{68} +x_{23} x_{63} +x_{22} x_{62} +x_{21} x_{61} +x_{15} x_{55} +x_{14} x_{54} +x_{8} x_{48} -16~\\x_{39} x_{79} +x_{36} x_{76} +x_{35} x_{75} +x_{31} x_{71} +x_{30} x_{70} +x_{29} x_{69} +x_{26} x_{66} +x_{23} x_{63} +x_{22} x_{62} +x_{21} x_{61} +x_{20} x_{60} +x_{18} x_{58} +x_{15} x_{55} +x_{14} x_{54} +x_{13} x_{53} +x_{12} x_{52} +x_{10} x_{50} +x_{8} x_{48} +x_{7} x_{47} +x_{6} x_{46} +x_{4} x_{44} +x_{3} x_{43} +x_{2} x_{42} +x_{1} x_{41} -24~\\x_{40} x_{80} +x_{39} x_{79} +x_{38} x_{78} +x_{36} x_{76} +x_{35} x_{75} +x_{34} x_{74} +2x_{31} x_{71} +x_{30} x_{70} +x_{29} x_{69} +x_{28} x_{68} +x_{27} x_{67} +2x_{23} x_{63} +x_{22} x_{62} +x_{21} x_{61} +x_{20} x_{60} +x_{19} x_{59} +2x_{15} x_{55} +x_{14} x_{54} +x_{13} x_{53} +x_{12} x_{52} +x_{11} x_{51} +2x_{8} x_{48} +x_{7} x_{47} +x_{6} x_{46} +x_{5} x_{45} +x_{3} x_{43} +x_{2} x_{42} +x_{1} x_{41} -32~\\x_{40} x_{80} +x_{39} x_{79} +x_{38} x_{78} +2x_{36} x_{76} +x_{35} x_{75} +x_{34} x_{74} +x_{33} x_{73} +2x_{31} x_{71} +2x_{30} x_{70} +x_{29} x_{69} +x_{28} x_{68} +x_{27} x_{67} +x_{26} x_{66} +x_{25} x_{65} +2x_{23} x_{63} +2x_{22} x_{62} +x_{21} x_{61} +x_{20} x_{60} +x_{19} x_{59} +x_{18} x_{58} +x_{17} x_{57} +2x_{15} x_{55} +2x_{14} x_{54} +2x_{13} x_{53} +x_{12} x_{52} +x_{11} x_{51} +x_{10} x_{50} +x_{9} x_{49} +2x_{8} x_{48} +2x_{7} x_{47} +x_{6} x_{46} +x_{5} x_{45} +x_{4} x_{44} +2x_{3} x_{43} +x_{2} x_{42} +2x_{1} x_{41} -48~\\x_{40} x_{80} +x_{39} x_{79} +x_{38} x_{78} +x_{37} x_{77} +x_{36} x_{76} +x_{35} x_{75} +x_{34} x_{74} +x_{33} x_{73} +x_{32} x_{72} +x_{31} x_{71} +x_{30} x_{70} +x_{29} x_{69} +x_{28} x_{68} +x_{27} x_{67} +x_{26} x_{66} +x_{25} x_{65} +x_{24} x_{64} +x_{23} x_{63} +x_{22} x_{62} +x_{21} x_{61} +x_{20} x_{60} +x_{19} x_{59} +x_{18} x_{58} +x_{17} x_{57} +x_{16} x_{56} +x_{15} x_{55} +x_{14} x_{54} +x_{13} x_{53} +x_{12} x_{52} +x_{11} x_{51} +x_{10} x_{50} +x_{9} x_{49} +x_{8} x_{48} +x_{7} x_{47} +x_{6} x_{46} +x_{5} x_{45} +x_{4} x_{44} +x_{3} x_{43} +x_{2} x_{42} +x_{1} x_{41} -40~\\x_{38} x_{78} +x_{35} x_{75} +x_{34} x_{74} +x_{32} x_{72} +x_{30} x_{70} +x_{29} x_{69} +x_{28} x_{68} +x_{25} x_{65} +x_{24} x_{64} +x_{23} x_{63} +x_{22} x_{62} +x_{21} x_{61} +x_{19} x_{59} +x_{18} x_{58} +x_{17} x_{57} +x_{16} x_{56} +x_{15} x_{55} +x_{14} x_{54} +x_{12} x_{52} +x_{11} x_{51} +x_{10} x_{50} +x_{9} x_{49} +x_{8} x_{48} +x_{7} x_{47} +x_{6} x_{46} +x_{5} x_{45} +x_{4} x_{44} +x_{3} x_{43} +x_{2} x_{42} +x_{1} x_{41} -30~\\x_{34} x_{74} +x_{29} x_{69} +x_{28} x_{68} +x_{24} x_{64} +x_{22} x_{62} +x_{21} x_{61} +x_{17} x_{57} +x_{16} x_{56} +x_{15} x_{55} +x_{14} x_{54} +x_{11} x_{51} +x_{10} x_{50} +x_{9} x_{49} +x_{8} x_{48} +x_{6} x_{46} +x_{5} x_{45} +x_{4} x_{44} +x_{3} x_{43} +x_{2} x_{42} +x_{1} x_{41} -20~\\x_{28} x_{68} +x_{21} x_{61} +x_{16} x_{56} +x_{14} x_{54} +x_{9} x_{49} +x_{8} x_{48} +x_{5} x_{45} +x_{4} x_{44} +x_{2} x_{42} +x_{1} x_{41} -10~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 1, 0, 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: 3
Containing regular semisimple subalgebra number 1: E^{1}_7 Containing regular semisimple subalgebra number 2: D^{1}_6+A^{1}_1 Containing regular semisimple subalgebra number 3: A^{1}_5+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+4V_{4\psi}+6V_{3\psi}+6V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+30h_{6}+39h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\)
\( e = 53530271443099815166337135775563458/70527034706578121197278255114098603g_{55}-37695270334867284807184542558940789/282108138826312484789113020456394412g_{50}+2529435709802163541605639035612849/282108138826312484789113020456394412g_{49}+403982936760680601109440610822161634/70527034706578121197278255114098603g_{47}+53783490387615/96578925814664g_{44}-306493155501535789917292386407003/70527034706578121197278255114098603g_{43}+1438696742660406129145827733279475/141054069413156242394556510228197206g_{42}-108635337563273869494679420764506770/70527034706578121197278255114098603g_{41}-43848467546421865210749641300333446/70527034706578121197278255114098603g_{40}-197192775990428034615333153940637454/70527034706578121197278255114098603g_{38}-5055497743173/48289462907332g_{37}+898773494620933763892917272411691/282108138826312484789113020456394412g_{36}+20751421562027643287498096514660334/70527034706578121197278255114098603g_{35}+14085912840802908453723529798976754/70527034706578121197278255114098603g_{34}+36401318079633079330892011217167003/70527034706578121197278255114098603g_{33}+19134593945095727448928735228099437/70527034706578121197278255114098603g_{32}+8397780608211/96578925814664g_{31}-1788447397522233845959347271382598/70527034706578121197278255114098603g_{28}-2509145483206436260214799982458976/70527034706578121197278255114098603g_{27}-2129546583572873413937488298874780/70527034706578121197278255114098603g_{26}-615592653714753891738169893121737/70527034706578121197278255114098603g_{20}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{19} x_{42} +x_{17} x_{41} +x_{14} x_{39} +x_{11} x_{37} +x_{10} x_{36} +x_{7} x_{34} +x_{5} x_{30} +x_{4} x_{29} +x_{1} x_{24} ~\\-x_{16} x_{41} -x_{11} x_{38} -x_{9} x_{36} -x_{5} x_{31} -x_{3} x_{29} -x_{1} x_{25} ~\\x_{13} x_{39} +x_{8} x_{36} +x_{7} x_{35} +x_{6} x_{33} +x_{4} x_{31} +x_{3} x_{30} +x_{1} x_{26} ~\\-x_{16} x_{42} -x_{11} x_{40} -x_{9} x_{39} -x_{5} x_{35} -x_{3} x_{34} -x_{1} x_{28} ~\\x_{13} x_{42} +x_{8} x_{41} +x_{7} x_{40} +x_{4} x_{38} +x_{3} x_{37} -x_{2} x_{33} +x_{1} x_{32} ~\\x_{18} x_{42} +x_{15} x_{41} +x_{14} x_{40} +x_{10} x_{38} +x_{9} x_{37} +x_{5} x_{32} +x_{2} x_{27} ~\\x_{21} x_{40} +x_{20} x_{38} +x_{18} x_{35} +x_{16} x_{32} +x_{15} x_{31} +x_{13} x_{28} +x_{9} x_{26} +x_{8} x_{25} +x_{3} x_{22} ~\\-x_{20} x_{37} -x_{17} x_{32} -x_{15} x_{30} -x_{10} x_{26} -x_{8} x_{24} -x_{4} x_{22} ~\\-x_{20} x_{42} -x_{17} x_{40} -x_{15} x_{39} -x_{10} x_{35} -x_{8} x_{34} -x_{4} x_{28} ~\\x_{18} x_{34} +x_{15} x_{29} +x_{14} x_{28} +x_{12} x_{27} +x_{10} x_{25} +x_{9} x_{24} +x_{5} x_{22} ~\\x_{21} x_{39} +x_{20} x_{36} +x_{19} x_{35} +x_{17} x_{31} +x_{16} x_{30} +x_{11} x_{26} +x_{6} x_{23} ~\\-x_{21} x_{37} -x_{19} x_{32} -x_{18} x_{30} -x_{14} x_{26} -x_{13} x_{24} -x_{7} x_{22} ~\\-x_{21} x_{41} -x_{19} x_{38} -x_{18} x_{36} -x_{14} x_{31} -x_{13} x_{29} -x_{7} x_{25} ~\\x_{21} x_{34} +x_{20} x_{29} +x_{19} x_{28} +x_{17} x_{25} +x_{16} x_{24} -x_{12} x_{23} +x_{11} x_{22} ~\\x_{13} x_{34} +x_{8} x_{29} +x_{7} x_{28} +x_{6} x_{27} +x_{4} x_{25} +x_{3} x_{24} +x_{2} x_{23} +x_{1} x_{22} -16~\\x_{19} x_{40} +x_{17} x_{38} +x_{14} x_{35} +x_{12} x_{33} +x_{11} x_{32} +x_{10} x_{31} +x_{7} x_{28} +x_{6} x_{27} +x_{5} x_{26} +x_{4} x_{25} +x_{2} x_{23} +x_{1} x_{22} -24~\\x_{18} x_{39} +x_{15} x_{36} +x_{14} x_{35} +x_{13} x_{34} +x_{12} x_{33} +x_{10} x_{31} +x_{9} x_{30} +x_{8} x_{29} +x_{7} x_{28} +x_{6} x_{27} +x_{5} x_{26} +x_{4} x_{25} +x_{3} x_{24} +2x_{2} x_{23} +x_{1} x_{22} -32~\\x_{21} x_{42} +x_{20} x_{41} +x_{19} x_{40} +x_{18} x_{39} +x_{17} x_{38} +x_{16} x_{37} +x_{15} x_{36} +x_{14} x_{35} +x_{13} x_{34} +2x_{12} x_{33} +x_{11} x_{32} +x_{10} x_{31} +x_{9} x_{30} +x_{8} x_{29} +x_{7} x_{28} +2x_{6} x_{27} +x_{5} x_{26} +x_{4} x_{25} +x_{3} x_{24} +2x_{2} x_{23} +x_{1} x_{22} -48~\\x_{21} x_{42} +x_{20} x_{41} +x_{19} x_{40} +x_{18} x_{39} +x_{17} x_{38} +x_{16} x_{37} +x_{15} x_{36} +x_{14} x_{35} +x_{13} x_{34} +x_{12} x_{33} +x_{11} x_{32} +x_{10} x_{31} +x_{9} x_{30} +x_{8} x_{29} +x_{7} x_{28} +x_{6} x_{27} +x_{5} x_{26} +x_{4} x_{25} +x_{3} x_{24} +x_{2} x_{23} +x_{1} x_{22} -39~\\x_{21} x_{42} +x_{20} x_{41} +x_{19} x_{40} +x_{18} x_{39} +x_{17} x_{38} +x_{16} x_{37} +x_{15} x_{36} +x_{14} x_{35} +x_{13} x_{34} +x_{11} x_{32} +x_{10} x_{31} +x_{9} x_{30} +x_{8} x_{29} +x_{7} x_{28} +x_{5} x_{26} +x_{4} x_{25} +x_{3} x_{24} +x_{1} x_{22} -30~\\x_{21} x_{42} +x_{19} x_{40} +x_{18} x_{39} +x_{16} x_{37} +x_{14} x_{35} +x_{13} x_{34} +x_{11} x_{32} +x_{9} x_{30} +x_{7} x_{28} +x_{5} x_{26} +x_{3} x_{24} +x_{1} x_{22} -20~\\x_{21} x_{42} +x_{19} x_{40} +x_{18} x_{39} +x_{14} x_{35} +x_{13} x_{34} +x_{7} x_{28} -10~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 0, 0, 1, 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: 2
Containing regular semisimple subalgebra number 1: D^{1}_4+A^{1}_3 Containing regular semisimple subalgebra number 2: D^{1}_6
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{10\psi}+4V_{9\psi}+V_{8\psi}+4V_{7\psi}+7V_{6\psi}+4V_{5\psi}+5V_{4\psi}+8V_{3\psi}+3V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+29h_{6}+38h_{5}+47h_{4}+32h_{3}+24h_{2}+16h_{1}\)
\( e = 6/5g_{46}+3/17g_{45}+2/13g_{43}+6g_{42}+5g_{40}+3/37g_{30}+3/5g_{17}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{5} x_{12} +x_{2} x_{9} -16~\\x_{7} x_{14} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -24~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} -32~\\x_{7} x_{14} +x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -47~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +2x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -38~\\x_{6} x_{13} +x_{5} x_{12} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -29~\\x_{6} x_{13} +x_{2} x_{9} +x_{1} x_{8} -20~\\x_{6} x_{13} +x_{1} x_{8} -10~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 1, 0, 1, 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: 2
Containing regular semisimple subalgebra number 1: E^{1}_6+A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_5+2A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{10\psi}+2V_{9\psi}+4V_{8\psi}+4V_{7\psi}+4V_{6\psi}+6V_{5\psi}+7V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}
+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+29h_{6}+38h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\)
\( e = -45887/66699g_{51}-7368532439494714/7820075754310791g_{50}+1/10g_{46}+34087/66699g_{45}+315603600211008/2606691918103597g_{43}+1098248817862130/7820075754310791g_{42}+8763094555186744/23460227262932373g_{41}+13557585053408/2606691918103597g_{36}+62482711656637/7820075754310791g_{35}+520694505747340/23460227262932373g_{34}-90412/535935g_{30}-126131946916723/7820075754310791g_{28}+72916/535935g_{24}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{11} x_{26} +x_{8} x_{25} +x_{6} x_{23} +x_{2} x_{20} +x_{1} x_{18} ~\\-x_{12} x_{26} -x_{8} x_{24} -x_{7} x_{23} -x_{2} x_{19} ~\\x_{10} x_{26} +x_{7} x_{25} +x_{6} x_{24} +x_{3} x_{22} +x_{2} x_{21} ~\\x_{13} x_{24} +x_{12} x_{21} +x_{10} x_{19} +x_{7} x_{15} +x_{5} x_{14} ~\\-x_{13} x_{25} -x_{11} x_{21} -x_{10} x_{20} -x_{6} x_{15} ~\\x_{13} x_{23} +x_{12} x_{20} +x_{11} x_{19} +x_{9} x_{16} +x_{8} x_{15} ~\\x_{9} x_{22} +x_{5} x_{18} +x_{3} x_{16} +x_{1} x_{14} -16~\\x_{10} x_{23} +x_{7} x_{20} +x_{6} x_{19} +x_{5} x_{18} +x_{4} x_{17} +x_{3} x_{16} +x_{2} x_{15} +x_{1} x_{14} -23~\\x_{11} x_{24} +x_{9} x_{22} +x_{8} x_{21} +x_{6} x_{19} +x_{5} x_{18} +x_{4} x_{17} +x_{3} x_{16} +x_{2} x_{15} +2x_{1} x_{14} -31~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{9} x_{22} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +2x_{5} x_{18} +2x_{4} x_{17} +x_{3} x_{16} +x_{2} x_{15} +2x_{1} x_{14} -46~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{9} x_{22} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +x_{5} x_{18} +2x_{4} x_{17} +x_{3} x_{16} +x_{2} x_{15} +x_{1} x_{14} -38~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +x_{5} x_{18} +x_{4} x_{17} +x_{2} x_{15} +x_{1} x_{14} -29~\\x_{13} x_{26} +x_{12} x_{25} +x_{11} x_{24} +x_{10} x_{23} +x_{8} x_{21} +x_{7} x_{20} +x_{6} x_{19} +x_{2} x_{15} -20~\\x_{12} x_{25} +x_{8} x_{21} +x_{7} x_{20} +x_{2} x_{15} -10~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 0, 2, 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: E^{1}_6 Containing regular semisimple subalgebra number 2: A^{1}_5+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{10\psi}+8V_{8\psi}+8V_{6\psi}+15V_{4\psi}+3V_{2\psi}+14V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+20h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\)
\( e = -14232/10811g_{69}+10072/10811g_{63}-7269341935/4204700608g_{50}+12068434175/37842305472g_{43}+272878515/4204700608g_{42}+2880303609/4204700608g_{41}+1035875005/37842305472g_{36}-2412096041/37842305472g_{35}+242415235/4204700608g_{34}-691636051/37842305472g_{28}+3256/9785g_{9}+248/1957g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{10} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{2} x_{18} +x_{1} x_{16} ~\\-x_{11} x_{24} -x_{7} x_{22} -x_{6} x_{21} -x_{2} x_{17} ~\\x_{9} x_{24} +x_{6} x_{23} +x_{5} x_{22} -x_{3} x_{20} +x_{2} x_{19} ~\\x_{12} x_{22} +x_{11} x_{19} +x_{9} x_{17} +x_{6} x_{14} +x_{4} x_{13} ~\\-x_{12} x_{23} -x_{10} x_{19} -x_{9} x_{18} -x_{5} x_{14} ~\\x_{12} x_{21} +x_{11} x_{18} +x_{10} x_{17} -x_{8} x_{15} +x_{7} x_{14} ~\\x_{8} x_{20} +x_{4} x_{16} +x_{3} x_{15} +x_{1} x_{13} -16~\\x_{10} x_{22} +x_{7} x_{19} +x_{5} x_{17} +x_{4} x_{16} +x_{2} x_{14} +2x_{1} x_{13} -22~\\x_{9} x_{21} +x_{6} x_{18} +x_{5} x_{17} +2x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} +2x_{1} x_{13} -30~\\x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +3x_{4} x_{16} +x_{2} x_{14} +3x_{1} x_{13} -44~\\x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +2x_{4} x_{16} +x_{2} x_{14} +2x_{1} x_{13} -36~\\x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{4} x_{16} +x_{2} x_{14} +x_{1} x_{13} -28~\\x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{2} x_{14} -20~\\x_{11} x_{23} +x_{7} x_{19} +x_{6} x_{18} +x_{2} x_{14} -10~\\\end{array}\)

h-characteristic: (1, 0, 0, 1, 0, 0, 0, 1)
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)
Containing regular semisimple subalgebra number 1: A^{1}_5+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+5V_{6\psi}+8V_{5\psi}+7V_{4\psi}+4V_{3\psi}+2V_{2\psi}+4V_{\psi}
+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+19h_{7}+28h_{6}+37h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\)
\( e = 5g_{50}+1/26g_{48}+9/5g_{46}+4g_{40}+5/17g_{36}+4/5g_{23}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{10} +x_{2} x_{8} -16~\\x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -23~\\x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -31~\\2x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -46~\\x_{6} x_{12} +x_{5} x_{11} +2x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -37~\\x_{6} x_{12} +x_{5} x_{11} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -28~\\x_{6} x_{12} +x_{5} x_{11} +x_{2} x_{8} +x_{1} x_{7} -19~\\x_{5} x_{11} +x_{1} x_{7} -10~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 1, 0, 1)
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: A^{1}_5
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{9\psi}+7V_{8\psi}+V_{6\psi}+14V_{5\psi}+7V_{4\psi}+2V_{3\psi}+V_{2\psi}+17V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\)
\( e = 5g_{62}+9/5g_{60}+4/5g_{24}+4g_{23}+5/17g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{9} +x_{2} x_{7} -16~\\x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -22~\\x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -30~\\x_{4} x_{9} +2x_{3} x_{8} +x_{2} x_{7} +2x_{1} x_{6} -44~\\x_{4} x_{9} +2x_{3} x_{8} +2x_{1} x_{6} -36~\\x_{5} x_{10} +2x_{3} x_{8} +x_{1} x_{6} -28~\\x_{5} x_{10} +x_{3} x_{8} +x_{1} x_{6} -19~\\x_{5} x_{10} +x_{1} x_{6} -10~\\\end{array}\)

h-characteristic: (0, 0, 1, 0, 0, 1, 0, 1)
Length of the weight dual to h: 68
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: D^{1}_5+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{9\psi}+3V_{8\psi}+6V_{7\psi}+4V_{6\psi}+6V_{5\psi}+6V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}
+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+15h_{1}\)
\( e = 2/17g_{52}+1/13g_{48}+853/595g_{47}+6g_{44}-3505/816g_{42}-2/595g_{40}+205/408g_{36}+451/336g_{33}-83/840g_{27}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{18} +x_{3} x_{17} +x_{2} x_{16} ~\\x_{9} x_{13} +x_{8} x_{12} +x_{7} x_{11} ~\\x_{8} x_{17} +x_{5} x_{14} +x_{3} x_{12} +x_{1} x_{10} -15~\\x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -22~\\x_{9} x_{18} +x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +2x_{1} x_{10} -30~\\x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +2x_{1} x_{10} -44~\\x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +2x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -36~\\x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} -28~\\x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{3} x_{12} +x_{2} x_{11} -19~\\x_{7} x_{16} +x_{2} x_{11} -10~\\\end{array}\)

h-characteristic: (0, 2, 0, 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: 3
Containing regular semisimple subalgebra number 1: D^{1}_4+4A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 3: D^{1}_4+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+6V_{8\psi}+14V_{6\psi}+7V_{4\psi}+14V_{2\psi}+8V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+22h_{2}+14h_{1}\)
\( e = 6g_{81}+1/17g_{53}+6/5g_{44}+1/50g_{41}+1/26g_{39}+1/37g_{38}+5g_{22}+3/5g_{18}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{15} +x_{5} x_{13} +x_{3} x_{11} +x_{1} x_{9} -14~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -22~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +2x_{3} x_{11} +2x_{1} x_{9} -28~\\x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +2x_{3} x_{11} +3x_{1} x_{9} -42~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +3x_{1} x_{9} -34~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{2} x_{10} +2x_{1} x_{9} -26~\\x_{8} x_{16} +x_{5} x_{13} +x_{2} x_{10} +x_{1} x_{9} -18~\\x_{2} x_{10} -10~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 0, 0, 2)
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: 2
Containing regular semisimple subalgebra number 1: D^{1}_4+3A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_5+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+3V_{8\psi}+6V_{7\psi}+8V_{6\psi}+6V_{5\psi}+3V_{4\psi}+2V_{3\psi}+7V_{2\psi}+10V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\)
\( e = 6g_{71}+1/26g_{54}+1/37g_{53}+6/5g_{52}+1/17g_{44}+3/5g_{39}+5g_{15}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -14~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -21~\\x_{7} x_{14} +x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{1} x_{8} -28~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{1} x_{8} -42~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +2x_{1} x_{8} -34~\\x_{7} x_{14} +x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +2x_{1} x_{8} -26~\\x_{7} x_{14} +x_{6} x_{13} +x_{2} x_{9} +x_{1} x_{8} -18~\\x_{2} x_{9} -10~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0, 1, 0, 2)
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: D^{1}_4+2A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_5
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+8V_{7\psi}+8V_{6\psi}+8V_{5\psi}+V_{4\psi}+8V_{2\psi}+8V_{\psi}+15V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+18h_{7}+26h_{6}+33h_{5}+40h_{4}+27h_{3}+20h_{2}+14h_{1}\)
\( e = 6g_{75}+1/17g_{63}+6/5g_{60}+3/5g_{40}+1/26g_{38}+5g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{1} x_{7} -14~\\x_{6} x_{12} +x_{5} x_{11} +x_{3} x_{9} +2x_{1} x_{7} -20~\\x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +2x_{1} x_{7} -27~\\x_{6} x_{12} +3x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +3x_{1} x_{7} -40~\\x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +2x_{1} x_{7} -33~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +x_{1} x_{7} -26~\\x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -18~\\x_{2} x_{8} -10~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 0, 1, 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)
Containing regular semisimple subalgebra number 1: A^{1}_4+A^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+6V_{6\psi}+6V_{5\psi}+6V_{4\psi}+8V_{3\psi}+6V_{2\psi}+4V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\)
\( e = 4g_{57}+3/17g_{43}+3g_{42}+3/37g_{41}+6/5g_{40}+2/5g_{39}+2/13g_{37}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{13} +x_{3} x_{10} +x_{1} x_{8} -14~\\x_{7} x_{14} +x_{6} x_{13} +x_{4} x_{11} +x_{2} x_{9} +x_{1} x_{8} -21~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{1} x_{8} -28~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +2x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -42~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -34~\\x_{7} x_{14} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -26~\\x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{2} x_{9} -18~\\x_{5} x_{12} +x_{2} x_{9} -9~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 0, 0, 1, 2)
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)
Containing regular semisimple subalgebra number 1: D^{1}_4+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+6V_{7\psi}+14V_{6\psi}+6V_{5\psi}+2V_{2\psi}+14V_{\psi}+21V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+18h_{7}+25h_{6}+32h_{5}+39h_{4}+26h_{3}+20h_{2}+13h_{1}\)
\( e = 6g_{81}+1/17g_{69}+6/5g_{58}+3/5g_{34}+5g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{10} +x_{3} x_{8} +x_{1} x_{6} -13~\\2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +x_{1} x_{6} -20~\\2x_{5} x_{10} +2x_{3} x_{8} +2x_{1} x_{6} -26~\\3x_{5} x_{10} +x_{4} x_{9} +2x_{3} x_{8} +3x_{1} x_{6} -39~\\2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +3x_{1} x_{6} -32~\\x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +2x_{1} x_{6} -25~\\x_{4} x_{9} +x_{3} x_{8} +x_{1} x_{6} -18~\\x_{2} x_{7} -10~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 2, 2)
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)
Containing regular semisimple subalgebra number 1: D^{1}_4
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+26V_{6\psi}+V_{2\psi}+52V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{8}+18h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\)
\( e = 6g_{97}+6/5g_{60}+5g_{8}+3/5g_{7}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -12~\\x_{3} x_{7} +2x_{1} x_{5} -18~\\x_{3} x_{7} +3x_{1} x_{5} -24~\\2x_{3} x_{7} +4x_{1} x_{5} -36~\\2x_{3} x_{7} +3x_{1} x_{5} -30~\\2x_{3} x_{7} +2x_{1} x_{5} -24~\\x_{4} x_{8} +x_{3} x_{7} +x_{1} x_{5} -18~\\x_{2} x_{6} -10~\\\end{array}\)

h-characteristic: (0, 0, 1, 0, 0, 1, 0, 0)
Length of the weight dual to h: 50
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}_4+A^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+10V_{4\psi}+8V_{3\psi}+7V_{2\psi}+6V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+24h_{6}+31h_{5}+38h_{4}+26h_{3}+19h_{2}+13h_{1}\)
\( e = 4g_{67}+3g_{48}+1/13g_{46}+1/37g_{44}+2/5g_{43}+2/17g_{40}+6/5g_{38}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{5} x_{12} +x_{3} x_{10} +x_{1} x_{8} -13~\\x_{7} x_{14} +x_{6} x_{13} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -19~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -26~\\2x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -38~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -31~\\x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -24~\\x_{5} x_{12} +x_{4} x_{11} +x_{2} x_{9} +x_{1} x_{8} -16~\\x_{4} x_{11} +x_{1} x_{8} -8~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 2, 0, 0)
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: 2
Containing regular semisimple subalgebra number 1: 2D^{1}_4 Containing regular semisimple subalgebra number 2: A^{1}_4+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+13V_{6\psi}+14V_{4\psi}+18V_{2\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+16h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\)
\( e = -21/4g_{82}+1/4g_{77}-965/1216g_{54}-11/736g_{52}+177/1216g_{48}+2g_{47}+31/736g_{45}+29/20g_{33}+477/874g_{29}-1/20g_{27}-73/874g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{3} x_{21} +x_{2} x_{20} +x_{1} x_{19} ~\\x_{6} x_{24} +x_{5} x_{23} +x_{4} x_{22} ~\\x_{9} x_{15} +x_{8} x_{14} +x_{7} x_{13} ~\\x_{12} x_{18} +x_{11} x_{17} +x_{10} x_{16} ~\\x_{11} x_{23} +x_{8} x_{20} +x_{7} x_{19} +x_{5} x_{17} +x_{2} x_{14} +x_{1} x_{13} -12~\\x_{11} x_{23} +x_{10} x_{22} +x_{7} x_{19} +x_{5} x_{17} +x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} +2x_{1} x_{13} -18~\\x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +2x_{7} x_{19} +x_{5} x_{17} +x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} +2x_{1} x_{13} -24~\\2x_{11} x_{23} +2x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +3x_{7} x_{19} +2x_{5} x_{17} +2x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} +3x_{1} x_{13} -36~\\x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +2x_{7} x_{19} +x_{6} x_{18} +2x_{5} x_{17} +2x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} +2x_{1} x_{13} -30~\\x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{4} x_{16} +x_{3} x_{15} +x_{2} x_{14} +x_{1} x_{13} -24~\\x_{12} x_{24} +x_{10} x_{22} +x_{8} x_{20} +x_{7} x_{19} +x_{6} x_{18} +x_{4} x_{16} +x_{2} x_{14} +x_{1} x_{13} -16~\\x_{12} x_{24} +x_{7} x_{19} +x_{6} x_{18} +x_{1} x_{13} -8~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 0, 0, 1)
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: 3
Containing regular semisimple subalgebra number 1: 2A^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_4+A^{1}_3 Containing regular semisimple subalgebra number 3: A^{1}_4+2A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{8\psi}+4V_{7\psi}+5V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+9V_{2\psi}+8V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+15h_{7}+22h_{6}+29h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\)
\( e = 3g_{71}+3/10g_{59}+1/50g_{48}+1/37g_{46}+3/26g_{45}+3/5g_{44}+4/17g_{43}+2g_{42}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -12~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -18~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +2x_{3} x_{11} +2x_{1} x_{9} -24~\\2x_{8} x_{16} +2x_{7} x_{15} +2x_{6} x_{14} +x_{5} x_{13} +2x_{4} x_{12} +2x_{3} x_{11} +x_{2} x_{10} +2x_{1} x_{9} -36~\\x_{8} x_{16} +2x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +2x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +2x_{1} x_{9} -29~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} +2x_{1} x_{9} -22~\\x_{8} x_{16} +x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} +x_{1} x_{9} -15~\\x_{5} x_{13} +x_{2} x_{10} -8~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0, 1, 0, 1)
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: 2
Containing regular semisimple subalgebra number 1: 2A^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_4+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{8\psi}+2V_{7\psi}+7V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+8V_{2\psi}+8V_{\psi}+9V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+15h_{7}+22h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\)
\( e = 3g_{75}+1/37g_{60}+3/5g_{52}+3/10g_{51}+2g_{43}+4/17g_{42}+3/26g_{40}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -12~\\x_{7} x_{14} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{1} x_{8} -17~\\x_{7} x_{14} +x_{6} x_{13} +2x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -23~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} +3x_{1} x_{8} -34~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -28~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -22~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{2} x_{9} +x_{1} x_{8} -15~\\x_{5} x_{12} +x_{2} x_{9} -8~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 0, 0, 0, 2)
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: 2A^{1}_3 Containing regular semisimple subalgebra number 2: A^{1}_4
sl(2)-module decomposition of the ambient Lie algebra: \(V_{8\psi}+11V_{6\psi}+21V_{4\psi}+11V_{2\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{8}+14h_{7}+20h_{6}+26h_{5}+32h_{4}+22h_{3}+16h_{2}+12h_{1}\)
\( e = 3g_{93}+3/10g_{52}+4/17g_{43}+2g_{42}+3/5g_{40}+3/26g_{23}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -12~\\x_{6} x_{12} +x_{4} x_{10} +x_{2} x_{8} +2x_{1} x_{7} -16~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +3x_{1} x_{7} -22~\\x_{6} x_{12} +x_{5} x_{11} +2x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +4x_{1} x_{7} -32~\\x_{5} x_{11} +2x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +3x_{1} x_{7} -26~\\x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -20~\\x_{5} x_{11} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -14~\\x_{5} x_{11} +x_{2} x_{8} -8~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 1, 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)
Containing regular semisimple subalgebra number 1: 2A^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(4V_{7\psi}+6V_{6\psi}+4V_{5\psi}+10V_{4\psi}+16V_{3\psi}+6V_{2\psi}+4V_{\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 7h_{8}+14h_{7}+21h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\)
\( e = 3g_{66}+4/17g_{62}+2g_{60}+3/10g_{47}+3/5g_{37}+3/26g_{32}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -12~\\x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -17~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -23~\\x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -34~\\x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +2x_{2} x_{8} +x_{1} x_{7} -28~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +2x_{2} x_{8} +x_{1} x_{7} -21~\\x_{5} x_{11} +x_{4} x_{10} +x_{2} x_{8} +x_{1} x_{7} -14~\\x_{5} x_{11} +x_{1} x_{7} -7~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 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)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: D^{1}_4+4A^{1}_1 Containing regular semisimple subalgebra number 2: 4A^{1}_2 Containing regular semisimple subalgebra number 3: A^{1}_3+A^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 4: D^{1}_4+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(8V_{6\psi}+20V_{4\psi}+28V_{2\psi}+8V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+16h_{2}+10h_{1}\)
\( e = 3511/2739g_{96}+1489/27390g_{92}+1/10g_{61}-2192/5313g_{51}+1/37g_{50}+1/17g_{48}+1/26g_{47}+1972/26565g_{44}+14815/13363g_{26}-503/26726g_{18}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{3} x_{20} -x_{2} x_{19} +x_{1} x_{18} ~\\-x_{10} x_{13} -x_{9} x_{12} +x_{8} x_{11} ~\\x_{9} x_{19} +x_{8} x_{18} +x_{6} x_{16} +x_{4} x_{14} +x_{2} x_{12} +x_{1} x_{11} -10~\\x_{10} x_{20} +x_{9} x_{19} +x_{8} x_{18} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +x_{3} x_{13} +x_{2} x_{12} +x_{1} x_{11} -16~\\2x_{9} x_{19} +2x_{8} x_{18} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +2x_{2} x_{12} +2x_{1} x_{11} -20~\\x_{10} x_{20} +2x_{9} x_{19} +3x_{8} x_{18} +x_{7} x_{17} +x_{6} x_{16} +2x_{5} x_{15} +2x_{4} x_{14} +x_{3} x_{13} +2x_{2} x_{12} +3x_{1} x_{11} -30~\\x_{10} x_{20} +x_{9} x_{19} +3x_{8} x_{18} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +x_{3} x_{13} +x_{2} x_{12} +3x_{1} x_{11} -24~\\2x_{8} x_{18} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +x_{3} x_{13} +x_{2} x_{12} +3x_{1} x_{11} -18~\\2x_{8} x_{18} +x_{7} x_{17} +x_{6} x_{16} +x_{5} x_{15} +x_{4} x_{14} +2x_{1} x_{11} -12~\\x_{8} x_{18} +x_{7} x_{17} +x_{4} x_{14} +x_{1} x_{11} -6~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0, 0, 0, 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: 2
Containing regular semisimple subalgebra number 1: D^{1}_4+3A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_3+A^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(5V_{6\psi}+6V_{5\psi}+10V_{4\psi}+14V_{3\psi}+15V_{2\psi}+10V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+15h_{2}+10h_{1}\)
\( e = 1028/775g_{84}+56/775g_{78}+1/17g_{62}+1/26g_{61}-181/372g_{59}+37/372g_{52}+847/750g_{48}+1/10g_{44}-19/750g_{39}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}-x_{3} x_{18} -x_{2} x_{17} +x_{1} x_{16} ~\\-x_{9} x_{12} -x_{8} x_{11} +x_{7} x_{10} ~\\x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{4} x_{13} +x_{2} x_{11} +x_{1} x_{10} -10~\\x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -15~\\x_{9} x_{18} +x_{8} x_{17} +2x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +2x_{1} x_{10} -20~\\2x_{9} x_{18} +2x_{8} x_{17} +2x_{7} x_{16} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} +2x_{2} x_{11} +2x_{1} x_{10} -30~\\x_{9} x_{18} +2x_{8} x_{17} +2x_{7} x_{16} +x_{6} x_{15} +2x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +2x_{2} x_{11} +2x_{1} x_{10} -24~\\x_{9} x_{18} +x_{8} x_{17} +2x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{3} x_{12} +x_{2} x_{11} +2x_{1} x_{10} -18~\\x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{3} x_{12} +x_{2} x_{11} +2x_{1} x_{10} -12~\\x_{7} x_{16} +x_{6} x_{15} +x_{5} x_{14} +x_{1} x_{10} -6~\\\end{array}\)

h-characteristic: (1, 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: 3
Containing regular semisimple subalgebra number 1: A^{1}_3+4A^{1}_1 Containing regular semisimple subalgebra number 2: D^{1}_4+2A^{1}_1 Containing regular semisimple subalgebra number 3: A^{1}_3+A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+8V_{5\psi}+8V_{4\psi}+16V_{3\psi}+16V_{2\psi}+8V_{\psi}+11V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+18h_{6}+23h_{5}+28h_{4}+19h_{3}+14h_{2}+10h_{1}\)
\( e = 3g_{82}+1/10g_{70}+2g_{60}+1/26g_{57}+3/5g_{49}+1/17g_{47}+1/37g_{45}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{1} x_{8} -10~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{2} x_{9} +2x_{1} x_{8} -14~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +2x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -19~\\2x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +3x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +3x_{1} x_{8} -28~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +2x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -23~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +x_{1} x_{8} -18~\\x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -12~\\x_{3} x_{10} +x_{1} x_{8} -6~\\\end{array}\)

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: 3A^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_3+3A^{1}_1 Containing regular semisimple subalgebra number 3: D^{1}_4+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{6\psi}+6V_{5\psi}+13V_{4\psi}+12V_{3\psi}+16V_{2\psi}+14V_{\psi}+9V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+17h_{6}+22h_{5}+27h_{4}+18h_{3}+14h_{2}+9h_{1}\)
\( e = 2g_{87}+1/37g_{69}+2/5g_{65}+g_{58}+1/5g_{56}+1/13g_{55}+2/17g_{54}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{6} x_{13} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -9~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -14~\\2x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -18~\\3x_{7} x_{14} +x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +3x_{1} x_{8} -27~\\2x_{7} x_{14} +x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} +3x_{1} x_{8} -22~\\x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} +2x_{1} x_{8} -17~\\x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -12~\\x_{6} x_{13} +x_{4} x_{11} +x_{1} x_{8} -6~\\\end{array}\)

h-characteristic: (0, 0, 1, 0, 0, 0, 0, 1)
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)
Containing regular semisimple subalgebra number 1: A^{1}_3+2A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+6V_{5\psi}+11V_{4\psi}+16V_{3\psi}+15V_{2\psi}+14V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+11h_{7}+16h_{6}+21h_{5}+26h_{4}+18h_{3}+13h_{2}+9h_{1}\)
\( e = 1/10g_{81}+3g_{79}+2g_{69}+1/17g_{58}+3/5g_{43}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{10} +x_{4} x_{9} +x_{2} x_{7} +x_{1} x_{6} -9~\\x_{5} x_{10} +x_{4} x_{9} +2x_{2} x_{7} +x_{1} x_{6} -13~\\2x_{5} x_{10} +2x_{4} x_{9} +x_{3} x_{8} +2x_{2} x_{7} +x_{1} x_{6} -18~\\2x_{5} x_{10} +3x_{4} x_{9} +x_{3} x_{8} +3x_{2} x_{7} +2x_{1} x_{6} -26~\\x_{5} x_{10} +3x_{4} x_{9} +x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -21~\\x_{5} x_{10} +2x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +2x_{1} x_{6} -16~\\x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +2x_{1} x_{6} -11~\\x_{3} x_{8} +x_{1} x_{6} -6~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 2, 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: 3A^{1}_2 Containing regular semisimple subalgebra number 2: A^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 3: D^{1}_4
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{6\psi}+25V_{4\psi}+27V_{2\psi}+28V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+12h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\)
\( e = 2g_{101}+2/5g_{75}+g_{60}+1/5g_{43}+1/13g_{42}+2/17g_{40}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{11} +x_{3} x_{9} +2x_{1} x_{7} -8~\\x_{6} x_{12} +2x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -12~\\x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +x_{2} x_{8} +3x_{1} x_{7} -16~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +3x_{3} x_{9} +2x_{2} x_{8} +4x_{1} x_{7} -24~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +3x_{1} x_{7} -20~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -16~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -12~\\x_{6} x_{12} +x_{4} x_{10} +x_{1} x_{7} -6~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 1, 0, 1)
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)
Containing regular semisimple subalgebra number 1: A^{1}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{5\psi}+15V_{4\psi}+18V_{3\psi}+10V_{2\psi}+14V_{\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+11h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\)
\( e = 3g_{82}+2g_{81}+1/10g_{80}+3/5g_{22}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{8} +x_{2} x_{6} +x_{1} x_{5} -8~\\2x_{4} x_{8} +x_{2} x_{6} +2x_{1} x_{5} -12~\\2x_{4} x_{8} +2x_{2} x_{6} +2x_{1} x_{5} -16~\\3x_{4} x_{8} +3x_{2} x_{6} +3x_{1} x_{5} -24~\\2x_{4} x_{8} +3x_{2} x_{6} +2x_{1} x_{5} -20~\\2x_{4} x_{8} +x_{3} x_{7} +2x_{2} x_{6} +x_{1} x_{5} -16~\\x_{4} x_{8} +x_{3} x_{7} +x_{2} x_{6} +x_{1} x_{5} -11~\\x_{3} x_{7} +x_{1} x_{5} -6~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0, 0, 0, 2)
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: A^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+11V_{4\psi}+32V_{3\psi}+V_{2\psi}+55V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\)
\( e = 2g_{97}+3g_{74}+3/5g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{5} -8~\\2x_{2} x_{5} +x_{1} x_{4} -11~\\3x_{2} x_{5} +x_{1} x_{4} -15~\\4x_{2} x_{5} +2x_{1} x_{4} -22~\\3x_{2} x_{5} +2x_{1} x_{4} -18~\\2x_{2} x_{5} +2x_{1} x_{4} -14~\\x_{2} x_{5} +2x_{1} x_{4} -10~\\x_{3} x_{6} +x_{1} x_{4} -6~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 1, 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: 2A^{1}_2+2A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(4V_{5\psi}+10V_{4\psi}+16V_{3\psi}+20V_{2\psi}+20V_{\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{8}+10h_{7}+15h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\)
\( e = 2g_{91}+2/5g_{78}+g_{63}+1/26g_{62}+1/17g_{60}+1/5g_{59}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -8~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -12~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -16~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +3x_{2} x_{8} +3x_{1} x_{7} -24~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -20~\\x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -15~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +2x_{1} x_{7} -10~\\x_{6} x_{12} +x_{3} x_{9} +x_{1} x_{7} -5~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0, 0, 1, 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: 2A^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{5\psi}+10V_{4\psi}+16V_{3\psi}+23V_{2\psi}+18V_{\psi}+17V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\)
\( e = 2g_{98}+2/5g_{75}+1/17g_{74}+g_{65}+1/5g_{49}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -8~\\x_{5} x_{10} +2x_{3} x_{8} +x_{2} x_{7} +2x_{1} x_{6} -11~\\x_{5} x_{10} +x_{4} x_{9} +2x_{3} x_{8} +x_{2} x_{7} +3x_{1} x_{6} -15~\\2x_{5} x_{10} +x_{4} x_{9} +3x_{3} x_{8} +2x_{2} x_{7} +4x_{1} x_{6} -22~\\2x_{5} x_{10} +x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +3x_{1} x_{6} -18~\\2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -14~\\2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -10~\\x_{5} x_{10} +x_{4} x_{9} +x_{1} x_{6} -5~\\\end{array}\)

h-characteristic: (2, 0, 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)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 8A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_2+4A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(14V_{4\psi}+50V_{2\psi}+28V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+8h_{1}\)
\( e = g_{112}+1/2g_{96}+1/5g_{77}+1/17g_{64}+1/10g_{55}+1/50g_{53}+1/26g_{52}+1/37g_{51}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -8~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +3x_{1} x_{9} -10~\\x_{8} x_{16} +2x_{7} x_{15} +x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +2x_{3} x_{11} +2x_{2} x_{10} +3x_{1} x_{9} -14~\\2x_{8} x_{16} +2x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +3x_{3} x_{11} +3x_{2} x_{10} +5x_{1} x_{9} -20~\\x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +2x_{3} x_{11} +3x_{2} x_{10} +4x_{1} x_{9} -16~\\x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +3x_{2} x_{10} +3x_{1} x_{9} -12~\\x_{8} x_{16} +x_{5} x_{13} +x_{4} x_{12} +x_{3} x_{11} +2x_{2} x_{10} +2x_{1} x_{9} -8~\\x_{4} x_{12} +x_{3} x_{11} +x_{2} x_{10} +x_{1} x_{9} -4~\\\end{array}\)

h-characteristic: (0, 0, 1, 0, 0, 0, 0, 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: 2
Containing regular semisimple subalgebra number 1: 7A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_2+3A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(7V_{4\psi}+14V_{3\psi}+28V_{2\psi}+28V_{\psi}+17V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+7h_{1}\)
\( e = g_{106}+1/2g_{91}+1/5g_{83}+1/10g_{72}+1/26g_{71}+1/37g_{70}+1/17g_{69}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -7~\\x_{7} x_{14} +x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -10~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -14~\\3x_{7} x_{14} +2x_{6} x_{13} +3x_{5} x_{12} +2x_{4} x_{11} +3x_{3} x_{10} +3x_{2} x_{9} +4x_{1} x_{8} -20~\\2x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +4x_{1} x_{8} -16~\\x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +3x_{1} x_{8} -12~\\x_{7} x_{14} +x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -8~\\x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} +x_{1} x_{8} -4~\\\end{array}\)

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: 2
Containing regular semisimple subalgebra number 1: 6A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_2+2A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+16V_{3\psi}+27V_{2\psi}+32V_{\psi}+24V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+8h_{7}+12h_{6}+15h_{5}+18h_{4}+12h_{3}+9h_{2}+6h_{1}\)
\( e = g_{104}+1/2g_{98}+1/17g_{89}+1/5g_{74}+1/10g_{73}+1/26g_{71}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{2} x_{8} +2x_{1} x_{7} -6~\\x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -9~\\2x_{6} x_{12} +2x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +3x_{2} x_{8} +3x_{1} x_{7} -12~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +4x_{2} x_{8} +4x_{1} x_{7} -18~\\2x_{6} x_{12} +3x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +3x_{2} x_{8} +3x_{1} x_{7} -15~\\2x_{6} x_{12} +2x_{5} x_{11} +2x_{4} x_{10} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -12~\\x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +2x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -8~\\x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0, 0, 0, 1)
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: 2
Containing regular semisimple subalgebra number 1: 5A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{4\psi}+12V_{3\psi}+32V_{2\psi}+32V_{\psi}+35V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+7h_{7}+10h_{6}+13h_{5}+16h_{4}+11h_{3}+8h_{2}+6h_{1}\)
\( e = g_{112}+1/17g_{97}+1/2g_{96}+1/5g_{77}+1/10g_{55}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -6~\\2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +3x_{1} x_{6} -8~\\3x_{5} x_{10} +x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +3x_{1} x_{6} -11~\\4x_{5} x_{10} +x_{4} x_{9} +3x_{3} x_{8} +3x_{2} x_{7} +5x_{1} x_{6} -16~\\3x_{5} x_{10} +x_{4} x_{9} +2x_{3} x_{8} +3x_{2} x_{7} +4x_{1} x_{6} -13~\\2x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +3x_{2} x_{7} +3x_{1} x_{6} -10~\\x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -7~\\x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -4~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 0, 2)
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: 2
Containing regular semisimple subalgebra number 1: 4A^{1}_1 Containing regular semisimple subalgebra number 2: A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{4\psi}+55V_{2\psi}+78V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\)
\( e = g_{119}+1/2g_{101}+1/5g_{68}+1/10g_{15}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{6} +2x_{1} x_{5} -4~\\x_{3} x_{7} +2x_{2} x_{6} +3x_{1} x_{5} -6~\\x_{3} x_{7} +3x_{2} x_{6} +4x_{1} x_{5} -8~\\2x_{3} x_{7} +4x_{2} x_{6} +6x_{1} x_{5} -12~\\2x_{3} x_{7} +3x_{2} x_{6} +5x_{1} x_{5} -10~\\2x_{3} x_{7} +2x_{2} x_{6} +4x_{1} x_{5} -8~\\x_{4} x_{8} +x_{3} x_{7} +x_{2} x_{6} +3x_{1} x_{5} -6~\\x_{4} x_{8} +x_{3} x_{7} +x_{2} x_{6} +x_{1} x_{5} -4~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 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)
Containing regular semisimple subalgebra number 1: 4A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(8V_{3\psi}+28V_{2\psi}+48V_{\psi}+36V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 3h_{8}+6h_{7}+9h_{6}+12h_{5}+15h_{4}+10h_{3}+8h_{2}+5h_{1}\)
\( e = g_{114}+1/2g_{106}+1/5g_{91}+1/10g_{69}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{8} +x_{3} x_{7} +x_{2} x_{6} +2x_{1} x_{5} -5~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +4x_{1} x_{5} -10~\\3x_{4} x_{8} +3x_{3} x_{7} +4x_{2} x_{6} +5x_{1} x_{5} -15~\\2x_{4} x_{8} +2x_{3} x_{7} +4x_{2} x_{6} +4x_{1} x_{5} -12~\\x_{4} x_{8} +2x_{3} x_{7} +3x_{2} x_{6} +3x_{1} x_{5} -9~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -6~\\x_{3} x_{7} +x_{2} x_{6} +x_{1} x_{5} -3~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 1, 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)
Containing regular semisimple subalgebra number 1: 3A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{3\psi}+27V_{2\psi}+52V_{\psi}+55V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 3h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\)
\( e = g_{118}+1/2g_{104}+1/5g_{74}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{5} +2x_{1} x_{4} -4~\\x_{3} x_{6} +2x_{2} x_{5} +3x_{1} x_{4} -6~\\x_{3} x_{6} +3x_{2} x_{5} +4x_{1} x_{4} -8~\\2x_{3} x_{6} +4x_{2} x_{5} +6x_{1} x_{4} -12~\\2x_{3} x_{6} +3x_{2} x_{5} +5x_{1} x_{4} -10~\\2x_{3} x_{6} +2x_{2} x_{5} +4x_{1} x_{4} -8~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\x_{3} x_{6} +x_{2} x_{5} +x_{1} x_{4} -3~\\\end{array}\)

h-characteristic: (1, 0, 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)
Containing regular semisimple subalgebra number 1: 2A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(14V_{2\psi}+64V_{\psi}+78V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 2h_{8}+4h_{7}+6h_{6}+8h_{5}+10h_{4}+7h_{3}+5h_{2}+4h_{1}\)
\( e = g_{120}+1/2g_{97}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +3x_{1} x_{3} -5~\\3x_{2} x_{4} +4x_{1} x_{3} -7~\\4x_{2} x_{4} +6x_{1} x_{3} -10~\\3x_{2} x_{4} +5x_{1} x_{3} -8~\\2x_{2} x_{4} +4x_{1} x_{3} -6~\\x_{2} x_{4} +3x_{1} x_{3} -4~\\2x_{1} x_{3} -2~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 0, 0, 0, 1)
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: \(V_{2\psi}+56V_{\psi}+133V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}\)
\( e = g_{120}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{2} -2~\\3x_{1} x_{2} -3~\\4x_{1} x_{2} -4~\\6x_{1} x_{2} -6~\\5x_{1} x_{2} -5~\\4x_{1} x_{2} -4~\\3x_{1} x_{2} -3~\\2x_{1} x_{2} -2~\\\end{array}\)