Subalgebra \(A^{8}_1+A^{3}_1+A^{1}_1\) ↪ \(E^{1}_6\)
65 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1\) .
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: \(\displaystyle A^{8}_1\): (4, 4, 6, 8, 6, 4): 16, \(\displaystyle A^{3}_1\): (0, 2, 2, 3, 2, 0): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}\), \(\displaystyle g_{-13}+g_{-14}-g_{-15}\), \(\displaystyle g_{-4}\)
Positive simple generators: \(\displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}\), \(\displaystyle -g_{15}+g_{14}+g_{13}\), \(\displaystyle g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\ 0 & 2/3 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
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 9) ; the vectors are over the primal subalgebra.\(g_{21}+g_{20}+g_{18}+g_{17}\)\(-g_{15}+g_{14}+g_{13}\)\(g_{4}\)\(g_{33}+g_{32}\)\(g_{31}+g_{29}\)\(-g_{28}+g_{26}\)\(g_{23}\)\(g_{36}\)\(g_{35}\)
weight\(2\omega_{1}\)\(2\omega_{2}\)\(2\omega_{3}\)\(4\omega_{1}\)\(2\omega_{1}+2\omega_{2}\)\(2\omega_{1}+\omega_{2}+\omega_{3}\)\(3\omega_{2}+\omega_{3}\)\(4\omega_{1}+2\omega_{2}\)\(4\omega_{1}+\omega_{2}+\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 9 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0)\(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2)\(\displaystyle V_{4\omega_{1}} \) → (4, 0, 0)\(\displaystyle V_{2\omega_{1}+2\omega_{2}} \) → (2, 2, 0)\(\displaystyle V_{2\omega_{1}+\omega_{2}+\omega_{3}} \) → (2, 1, 1)\(\displaystyle V_{3\omega_{2}+\omega_{3}} \) → (0, 3, 1)\(\displaystyle V_{4\omega_{1}+2\omega_{2}} \) → (4, 2, 0)\(\displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}} \) → (4, 1, 1)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)\(W_{8}\)\(W_{9}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{21}-g_{20}-g_{18}-g_{17}\)
\(2h_{6}+3h_{5}+4h_{4}+3h_{3}+2h_{2}+2h_{1}\)
\(g_{-17}+g_{-18}+g_{-20}+g_{-21}\)
Semisimple subalgebra component.
\(g_{15}-g_{14}-g_{13}\)
\(2h_{5}+3h_{4}+2h_{3}+2h_{2}\)
\(2g_{-13}+2g_{-14}-2g_{-15}\)
Semisimple subalgebra component.
\(-g_{4}\)
\(h_{4}\)
\(2g_{-4}\)
\(g_{33}+g_{32}\)
\(g_{21}+g_{20}-g_{18}-g_{17}\)
\(-2h_{6}-h_{5}+h_{3}+2h_{1}\)
\(3g_{-17}+3g_{-18}-3g_{-20}-3g_{-21}\)
\(6g_{-32}+6g_{-33}\)
\(g_{31}+g_{29}\)
\(-g_{14}+g_{13}\)
\(g_{21}-g_{20}-g_{18}+g_{17}\)
\(-g_{-1}-g_{-6}\)
\(h_{5}-h_{3}\)
\(-2g_{6}-2g_{1}\)
\(-g_{-17}+g_{-18}+g_{-20}-g_{-21}\)
\(-2g_{-13}+2g_{-14}\)
\(2g_{-29}+2g_{-31}\)
\(-g_{28}+g_{26}\)
\(g_{10}-g_{9}\)
\(-g_{16}+g_{12}\)
\(-g_{25}+g_{22}\)
\(-g_{-7}-g_{-11}\)
\(g_{-3}+g_{-5}\)
\(g_{5}+g_{3}\)
\(-g_{11}-g_{7}\)
\(-g_{-22}+g_{-25}\)
\(-g_{-12}+g_{-16}\)
\(g_{-9}-g_{-10}\)
\(g_{-26}-g_{-28}\)
\(g_{23}\)
\(g_{10}+g_{9}-g_{8}\)
\(g_{19}\)
\(-2g_{-2}+2g_{-3}-2g_{-5}\)
\(g_{5}-g_{3}+g_{2}\)
\(6g_{-19}\)
\(-2g_{-8}+2g_{-9}+2g_{-10}\)
\(-6g_{-23}\)
\(g_{36}\)
\(g_{31}-g_{29}\)
\(-g_{33}+g_{32}\)
\(-2g_{15}-g_{14}-g_{13}\)
\(g_{21}-g_{20}+g_{18}-g_{17}\)
\(-2g_{24}\)
\(3g_{-1}-3g_{-6}\)
\(-h_{5}-h_{3}+2h_{2}\)
\(-2g_{6}+2g_{1}\)
\(6g_{-24}\)
\(3g_{-17}-3g_{-18}+3g_{-20}-3g_{-21}\)
\(2g_{-13}+2g_{-14}+4g_{-15}\)
\(6g_{-32}-6g_{-33}\)
\(-6g_{-29}+6g_{-31}\)
\(-12g_{-36}\)
\(g_{35}\)
\(g_{28}+g_{26}\)
\(g_{30}\)
\(g_{34}\)
\(g_{10}+g_{9}+2g_{8}\)
\(g_{16}+g_{12}\)
\(g_{25}+g_{22}\)
\(g_{27}\)
\(-3g_{-7}+3g_{-11}\)
\(-2g_{-2}-g_{-3}+g_{-5}\)
\(g_{5}-g_{3}-2g_{2}\)
\(g_{11}-g_{7}\)
\(-6g_{-27}\)
\(-3g_{-22}-3g_{-25}\)
\(-3g_{-12}-3g_{-16}\)
\(-2g_{-8}-g_{-9}-g_{-10}\)
\(6g_{-34}\)
\(6g_{-30}\)
\(3g_{-26}+3g_{-28}\)
\(-6g_{-35}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(2\omega_{1}+2\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-2\omega_{1}-2\omega_{2}\)		\(2\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(2\omega_{1}-\omega_{2}+\omega_{3}\)
\(2\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{2}+\omega_{3}\)
\(-2\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{2}-\omega_{3}\)		\(3\omega_{2}+\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(3\omega_{2}-\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-3\omega_{2}+\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-3\omega_{2}-\omega_{3}\)		\(4\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-2\omega_{2}\)		\(4\omega_{1}+\omega_{2}+\omega_{3}\)
\(2\omega_{1}+\omega_{2}+\omega_{3}\)
\(4\omega_{1}-\omega_{2}+\omega_{3}\)
\(4\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(2\omega_{1}-\omega_{2}+\omega_{3}\)
\(2\omega_{1}+\omega_{2}-\omega_{3}\)
\(4\omega_{1}-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}-\omega_{3}\)
\(-4\omega_{1}+\omega_{2}+\omega_{3}\)
\(-2\omega_{1}-\omega_{2}+\omega_{3}\)
\(-2\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-4\omega_{1}-\omega_{2}+\omega_{3}\)
\(-4\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{2}-\omega_{3}\)
\(-4\omega_{1}-\omega_{2}-\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(2\omega_{1}+2\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-2\omega_{1}-2\omega_{2}\)		\(2\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(2\omega_{1}-\omega_{2}+\omega_{3}\)
\(2\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{2}+\omega_{3}\)
\(-2\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{2}-\omega_{3}\)		\(3\omega_{2}+\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(3\omega_{2}-\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-3\omega_{2}+\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-3\omega_{2}-\omega_{3}\)		\(4\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-2\omega_{2}\)		\(4\omega_{1}+\omega_{2}+\omega_{3}\)
\(2\omega_{1}+\omega_{2}+\omega_{3}\)
\(4\omega_{1}-\omega_{2}+\omega_{3}\)
\(4\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(2\omega_{1}-\omega_{2}+\omega_{3}\)
\(2\omega_{1}+\omega_{2}-\omega_{3}\)
\(4\omega_{1}-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}+\omega_{2}+\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}-\omega_{3}\)
\(-4\omega_{1}+\omega_{2}+\omega_{3}\)
\(-2\omega_{1}-\omega_{2}+\omega_{3}\)
\(-2\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-4\omega_{1}-\omega_{2}+\omega_{3}\)
\(-4\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{2}-\omega_{3}\)
\(-4\omega_{1}-\omega_{2}-\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{2\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{-2\omega_{1}-2\omega_{2}}\)		\(\displaystyle M_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}-\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{3\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{3\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-3\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{-3\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}
\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}\)		\(\displaystyle M_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{4\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{4\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{4\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-4\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-2\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{-4\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-4\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-2\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-4\omega_{1}-\omega_{2}-\omega_{3}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{2\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{-2\omega_{1}-2\omega_{2}}\)		\(\displaystyle M_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}-\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{3\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{3\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-3\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{-3\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}
\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}\)		\(\displaystyle M_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{4\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{4\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{4\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-4\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-2\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{-4\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-4\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-2\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-4\omega_{1}-\omega_{2}-\omega_{3}}\)

Semisimple subalgebra: W_{1}+W_{2}+W_{3}
Centralizer extension: 0

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.
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Made total 925 arithmetic operations while solving the Serre relations polynomial system.