Subalgebra \(A^{84}_1\) ↪ \(E^{1}_6\)
19 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{84}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle E^{1}_6\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{84}_1\): (12, 16, 22, 30, 22, 12): 168
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-6}+g_{-9}+g_{-10}\)
Positive simple generators: \(\displaystyle 22g_{10}+8g_{9}-8g_{8}+12g_{6}+14g_{5}+14g_{3}+16g_{2}+12g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/42\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}168\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{16\omega_{1}}\oplus V_{14\omega_{1}}\oplus 2V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\)
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra.
Highest vectors of representations (total 8) ; the vectors are over the primal subalgebra.\(g_{10}+4/11g_{9}-4/11g_{8}+6/11g_{6}+7/11g_{5}+7/11g_{3}+8/11g_{2}+6/11g_{1}\)\(g_{16}+35/6g_{15}-10/3g_{14}+10/3g_{13}-6g_{12}+7g_{11}+7g_{7}\)\(g_{23}+7/10g_{21}-4/5g_{20}+7/10g_{18}+4/5g_{17}\)\(-g_{28}-4/11g_{26}-7/11g_{25}+21/22g_{24}+7/11g_{22}\)\(g_{30}+35/22g_{29}+7/11g_{27}\)\(g_{31}+4/11g_{29}+6/11g_{27}\)\(g_{35}+2g_{34}\)\(g_{36}\)
weight\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)\(10\omega_{1}\)\(14\omega_{1}\)\(16\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 7 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{4\omega_{1}} \) → (4)\(\displaystyle V_{6\omega_{1}} \) → (6)\(\displaystyle V_{8\omega_{1}} \) → (8)\(\displaystyle V_{10\omega_{1}} \) → (10)\(\displaystyle V_{14\omega_{1}} \) → (14)\(\displaystyle V_{16\omega_{1}} \) → (16)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-11/6g_{10}-2/3g_{9}+2/3g_{8}-g_{6}-7/6g_{5}-7/6g_{3}-4/3g_{2}-g_{1}\)
\(h_{6}+11/6h_{5}+5/2h_{4}+11/6h_{3}+4/3h_{2}+h_{1}\)
\(1/6g_{-1}+1/6g_{-2}+1/6g_{-3}+1/6g_{-6}+1/6g_{-9}+1/6g_{-10}\)
\(g_{16}+35/6g_{15}-10/3g_{14}+10/3g_{13}-6g_{12}+7g_{11}+7g_{7}\)
\(3/2g_{10}-8/3g_{9}+10/3g_{8}+g_{6}-7/6g_{5}+7/6g_{3}-g_{1}\)
\(2/3g_{4}-h_{6}-3/2h_{5}+7/6h_{4}+3/2h_{3}+h_{1}+7/3g_{-4}\)
\(1/2g_{-1}-7/6g_{-2}+1/6g_{-3}+2/3g_{-5}-1/2g_{-6}-7/3g_{-8}+5/6g_{-9}-5/6g_{-10}\)
\(1/3g_{-7}+2/3g_{-11}-1/3g_{-12}+1/3g_{-13}-1/3g_{-14}+1/3g_{-15}-1/3g_{-16}\)
\(g_{23}+7/10g_{21}-4/5g_{20}+7/10g_{18}+4/5g_{17}\)
\(-1/10g_{16}-1/5g_{14}-1/5g_{13}+4/5g_{12}+7/10g_{11}-7/10g_{7}\)
\(-1/10g_{10}+3/5g_{9}-1/5g_{8}-1/10g_{6}-7/10g_{5}-7/10g_{3}+2/5g_{2}-1/10g_{1}\)
\(2/5g_{4}+1/10h_{6}+1/10h_{5}-1/2h_{4}+1/10h_{3}-2/5h_{2}+1/10h_{1}\)
\(1/10g_{-1}-3/10g_{-2}+1/5g_{-3}+2/5g_{-5}+1/10g_{-6}-1/5g_{-9}-1/5g_{-10}\)
\(-1/10g_{-7}+2/5g_{-11}+3/10g_{-12}-1/10g_{-13}-1/10g_{-14}-3/10g_{-16}\)
\(-1/5g_{-17}-1/10g_{-18}+1/10g_{-19}+1/5g_{-20}-1/10g_{-21}-1/5g_{-23}\)
\(-g_{28}-4/11g_{26}-7/11g_{25}+21/22g_{24}+7/11g_{22}\)
\(7/11g_{23}+7/22g_{21}+4/11g_{20}+14/11g_{19}-7/22g_{18}+4/11g_{17}\)
\(15/22g_{16}+7/11g_{15}+3/11g_{14}-3/11g_{13}+4/11g_{12}+7/22g_{11}+7/22g_{7}\)
\(5/22g_{10}+1/11g_{9}-3/11g_{8}+15/22g_{6}+7/22g_{5}-7/22g_{3}-15/22g_{1}\)
\(-2/11g_{4}-15/22h_{6}-5/22h_{5}-7/22h_{4}+5/22h_{3}+15/22h_{1}-7/11g_{-4}\)
\(25/22g_{-1}+7/22g_{-2}+3/11g_{-3}-2/11g_{-5}-25/22g_{-6}+7/11g_{-8}+1/11g_{-9}-1/11g_{-10}\)
\(19/22g_{-7}-2/11g_{-11}+23/22g_{-12}-9/22g_{-13}+9/22g_{-14}+6/11g_{-15}+23/22g_{-16}\)
\(-7/11g_{-17}+7/22g_{-18}-21/22g_{-19}-7/11g_{-20}-7/22g_{-21}\)
\(7/11g_{-22}+7/11g_{-24}-7/11g_{-26}-7/11g_{-28}\)
\(g_{30}+35/22g_{29}+7/11g_{27}\)
\(g_{28}+13/22g_{26}+7/11g_{25}+7/11g_{24}+21/22g_{22}\)
\(-9/22g_{23}+14/11g_{21}-4/11g_{20}+7/22g_{19}+7/22g_{18}-13/22g_{17}\)
\(10/11g_{16}-7/11g_{15}+12/11g_{14}-2/11g_{13}-13/22g_{12}+14/11g_{11}-7/22g_{7}\)
\(-5/11g_{10}-17/22g_{9}-2/11g_{8}+10/11g_{6}-21/11g_{5}+7/22g_{3}-10/11g_{2}+10/11g_{1}\)
\(-21/22g_{4}-10/11h_{6}+5/11h_{5}+27/22h_{4}+5/11h_{3}+10/11h_{2}-10/11h_{1}+49/22g_{-4}\)
\(-25/11g_{-1}+13/22g_{-2}+17/11g_{-3}-21/22g_{-5}-25/11g_{-6}-49/22g_{-8}-g_{-9}+27/22g_{-10}\)
\(-42/11g_{-7}-21/22g_{-11}-14/11g_{-12}+42/11g_{-13}-7/11g_{-14}+14/11g_{-15}+7/2g_{-16}\)
\(-28/11g_{-17}-56/11g_{-18}-7/11g_{-19}-91/22g_{-20}-14/11g_{-21}+35/11g_{-23}\)
\(63/11g_{-22}-42/11g_{-24}+105/22g_{-25}-63/11g_{-26}-21/22g_{-28}\)
\(105/22g_{-27}-105/22g_{-30}+105/22g_{-31}\)
\(g_{31}+4/11g_{29}+6/11g_{27}\)
\(4/11g_{26}-5/11g_{25}+6/11g_{24}-2/11g_{22}\)
\(4/11g_{23}+1/11g_{21}-5/11g_{20}+3/11g_{19}-8/11g_{18}-4/11g_{17}\)
\(-4/11g_{16}-6/11g_{15}+4/11g_{14}-8/11g_{13}-4/11g_{12}+1/11g_{11}+8/11g_{7}\)
\(2/11g_{10}-12/11g_{9}-8/11g_{8}-4/11g_{6}-7/11g_{5}+14/11g_{3}+4/11g_{2}-4/11g_{1}\)
\(-20/11g_{4}+4/11h_{6}-2/11h_{5}+10/11h_{4}-2/11h_{3}-4/11h_{2}+4/11h_{1}+21/11g_{-4}\)
\(10/11g_{-1}-18/11g_{-2}+2/11g_{-3}-20/11g_{-5}+10/11g_{-6}-21/11g_{-8}-g_{-9}+10/11g_{-10}\)
\(8/11g_{-7}-20/11g_{-11}+21/11g_{-12}+14/11g_{-13}-28/11g_{-14}+12/11g_{-15}\)
\(-35/11g_{-17}-4/11g_{-18}+16/11g_{-19}-28/11g_{-20}+32/11g_{-21}-14/11g_{-23}\)
\(-12/11g_{-22}-36/11g_{-24}+12/11g_{-25}-21/11g_{-26}-42/11g_{-28}\)
\(12/11g_{-27}+3g_{-29}+21/11g_{-30}+12/11g_{-31}\)
\(g_{35}+2g_{34}\)
\(g_{33}+g_{32}\)
\(g_{31}+g_{30}-g_{29}\)
\(g_{28}-2g_{26}-g_{25}-g_{22}\)
\(-3g_{23}-g_{21}-2g_{20}-g_{18}+2g_{17}\)
\(-3g_{16}+5g_{14}+5g_{13}+2g_{12}-g_{11}+g_{7}\)
\(8g_{10}+7g_{9}+5g_{8}-3g_{6}+g_{5}+g_{3}-10g_{2}-3g_{1}\)
\(12g_{4}+3h_{6}-8h_{5}-15h_{4}-8h_{3}+10h_{2}+3h_{1}\)
\(14g_{-1}+35g_{-2}-16g_{-3}+12g_{-5}+14g_{-6}-28g_{-9}-28g_{-10}\)
\(30g_{-7}+12g_{-11}+42g_{-12}+63g_{-13}+63g_{-14}-42g_{-16}\)
\(-105g_{-17}+30g_{-18}-63g_{-19}+105g_{-20}+30g_{-21}+126g_{-23}\)
\(33g_{-22}-198g_{-25}-231g_{-26}+231g_{-28}\)
\(231g_{-27}+198g_{-29}-462g_{-30}-198g_{-31}\)
\(429g_{-32}+429g_{-33}\)
\(-429g_{-34}\)
\(g_{36}\)
\(g_{35}\)
\(-g_{33}+g_{32}\)
\(-g_{31}+g_{30}-g_{29}+2g_{27}\)
\(g_{28}-2g_{26}+3g_{25}+2g_{24}-3g_{22}\)
\(-3g_{23}+5g_{21}+2g_{20}-6g_{19}-5g_{18}+2g_{17}\)
\(7g_{16}-16g_{15}-5g_{14}+5g_{13}+2g_{12}+5g_{11}+5g_{7}\)
\(-28g_{10}+7g_{9}+5g_{8}+7g_{6}-21g_{5}+21g_{3}-7g_{1}\)
\(12g_{4}-7h_{6}+28h_{5}+21h_{4}-28h_{3}+7h_{1}+42g_{-4}\)
\(42g_{-1}-21g_{-2}-96g_{-3}+12g_{-5}-42g_{-6}-42g_{-8}-84g_{-9}+84g_{-10}\)
\(138g_{-7}+12g_{-11}+126g_{-12}+105g_{-13}-105g_{-14}-192g_{-15}+126g_{-16}\)
\(-231g_{-17}+330g_{-18}+297g_{-19}-231g_{-20}-330g_{-21}\)
\(-627g_{-22}+660g_{-24}+858g_{-25}-231g_{-26}-231g_{-28}\)
\(-2145g_{-27}+858g_{-29}+858g_{-31}\)
\(3003g_{-32}-3003g_{-33}\)
\(3003g_{-34}-6006g_{-35}\)
\(6006g_{-36}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)		\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)		\(16\omega_{1}\)
\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
\(-16\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)		\(\displaystyle M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}
\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}
\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle 2M_{10\omega_{1}}\oplus 2M_{8\omega_{1}}\oplus 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}
\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\oplus 2M_{-8\omega_{1}}\oplus 2M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)		\(\displaystyle M_{16\omega_{1}}\oplus M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}
\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}
\oplus M_{-12\omega_{1}}\oplus M_{-14\omega_{1}}\oplus M_{-16\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 3582365 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
1*2 (unknown) gens:
(
x_{1} g_{-1}+x_{2} g_{-2}+x_{3} g_{-3}+x_{4} g_{-5}+x_{5} g_{-6}+x_{6} g_{-8}+x_{7} g_{-9}+x_{8} g_{-10}, x_{16} g_{10}+x_{15} g_{9}+x_{14} g_{8}+x_{13} g_{6}+x_{12} g_{5}+x_{11} g_{3}+x_{10} g_{2}+x_{9} g_{1})
h: (12, 16, 22, 30, 22, 12), e = combination of g_{1} g_{2} g_{3} g_{5} g_{6} g_{8} g_{9} g_{10} , f= combination of g_{-1} g_{-2} g_{-3} g_{-5} g_{-6} g_{-8} g_{-9} g_{-10} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{16\omega_{1}}+2V_{14\omega_{1}}+2V_{12\omega_{1}}+4V_{10\omega_{1}}+5V_{8\omega_{1}}+6V_{6\omega_{1}}+7V_{4\omega_{1}}+8V_{2\omega_{1}}+8V_{0}+8V_{-2\omega_{1}}+7V_{-4\omega_{1}}+6V_{-6\omega_{1}}+5V_{-8\omega_{1}}+4V_{-10\omega_{1}}+2V_{-12\omega_{1}}+2V_{-14\omega_{1}}+V_{-16\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{9} -12= 0
x_{6} x_{14} +x_{2} x_{10} -16= 0
x_{8} x_{12} -x_{7} x_{11} -x_{6} x_{10} = 0
x_{7} x_{15} +x_{3} x_{11} -22= 0
x_{8} x_{16} +x_{4} x_{12} -22= 0
x_{5} x_{13} -12= 0
x_{4} x_{16} -x_{3} x_{15} -x_{2} x_{14} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} -30= 0
The above system after transformation.
x_{1} x_{9} -12= 0
x_{6} x_{14} +x_{2} x_{10} -16= 0
x_{8} x_{12} -x_{7} x_{11} -x_{6} x_{10} = 0
x_{7} x_{15} +x_{3} x_{11} -22= 0
x_{8} x_{16} +x_{4} x_{12} -22= 0
x_{5} x_{13} -12= 0
x_{4} x_{16} -x_{3} x_{15} -x_{2} x_{14} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} -30= 0
For the calculator:
(DynkinType =A^{84}_1; ElementsCartan =((12, 16, 22, 30, 22, 12)); generators =(x_{1} g_{-1}+x_{2} g_{-2}+x_{3} g_{-3}+x_{4} g_{-5}+x_{5} g_{-6}+x_{6} g_{-8}+x_{7} g_{-9}+x_{8} g_{-10}, x_{16} g_{10}+x_{15} g_{9}+x_{14} g_{8}+x_{13} g_{6}+x_{12} g_{5}+x_{11} g_{3}+x_{10} g_{2}+x_{9} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{9} -12, x_{6} x_{14} +x_{2} x_{10} -16, x_{8} x_{12} -x_{7} x_{11} -x_{6} x_{10} , x_{7} x_{15} +x_{3} x_{11} -22, x_{8} x_{16} +x_{4} x_{12} -22, x_{5} x_{13} -12, x_{4} x_{16} -x_{3} x_{15} -x_{2} x_{14} , x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} -30 )