Subalgebra \(A^{2}_2+A^{1}_2\) ↪ \(E^{1}_6\)
99 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{1}_1\) .
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: \(\displaystyle A^{2}_2\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \(\displaystyle A^{1}_2\): (0, 1, 0, 1, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{-8}\), \(\displaystyle g_{4}\)
Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{8}\), \(\displaystyle g_{-4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & 0 & 0\\ 0 & 0 & 2 & -1\\ 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 2 & -1\\ 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\)
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 7) ; the vectors are over the primal subalgebra.\(g_{31}\)\(g_{29}\)\(g_{14}+g_{13}\)\(g_{25}+g_{22}\)\(g_{2}\)\(g_{36}\)\(g_{19}\)
weight\(\omega_{1}+\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(\omega_{2}+\omega_{3}\)\(\omega_{1}+\omega_{4}\)\(\omega_{3}+\omega_{4}\)\(2\omega_{1}+\omega_{3}\)\(2\omega_{2}+\omega_{4}\)
Isotypic module decomposition over primal subalgebra (total 7 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}+\omega_{2}} \) → (1, 1, 0, 0)\(\displaystyle V_{\omega_{2}+\omega_{3}} \) → (0, 1, 1, 0)\(\displaystyle V_{\omega_{1}+\omega_{4}} \) → (1, 0, 0, 1)\(\displaystyle V_{\omega_{3}+\omega_{4}} \) → (0, 0, 1, 1)\(\displaystyle V_{2\omega_{1}+\omega_{3}} \) → (2, 0, 1, 0)\(\displaystyle V_{2\omega_{2}+\omega_{4}} \) → (0, 2, 0, 1)
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.
\(g_{31}-g_{29}\)
\(-g_{-1}-g_{-6}\)
\(g_{33}+g_{32}\)
\(-h_{6}-h_{1}\)
\(-2h_{6}-3h_{5}-4h_{4}-3h_{3}-2h_{2}-2h_{1}\)
\(-g_{-32}-g_{-33}\)
\(2g_{6}+2g_{1}\)
\(-g_{-29}+g_{-31}\)
\(g_{31}\)
\(-g_{-1}\)
\(g_{33}\)
\(-h_{1}\)
\(-h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-h_{1}\)
\(-g_{-33}\)
\(2g_{1}\)
\(g_{-31}\)
\(g_{14}+g_{13}\)
\(-g_{20}+g_{17}\)
\(g_{5}-g_{3}\)
\(g_{-18}+g_{-21}\)
\(-g_{11}-g_{7}\)
\(g_{10}+g_{9}\)
\(g_{-26}+g_{-28}\)
\(-g_{16}+g_{12}\)
\(-g_{-22}-g_{-25}\)
\(g_{25}+g_{22}\)
\(g_{-12}-g_{-16}\)
\(g_{28}+g_{26}\)
\(-g_{-9}-g_{-10}\)
\(g_{-7}+g_{-11}\)
\(-g_{21}-g_{18}\)
\(-g_{-3}+g_{-5}\)
\(g_{-17}-g_{-20}\)
\(-g_{-13}-g_{-14}\)
Semisimple subalgebra component.
\(-g_{2}\)
\(-g_{-4}\)
\(g_{8}\)
\(-h_{4}\)
\(-h_{4}-h_{2}\)
\(-g_{-8}\)
\(2g_{4}\)
\(-g_{-2}\)
\(g_{36}\)
\(-g_{14}+g_{13}\)
\(g_{34}\)
\(-2g_{-24}\)
\(g_{20}+g_{17}\)
\(-g_{5}-g_{3}\)
\(g_{35}\)
\(-2g_{-18}+2g_{-21}\)
\(-2g_{-30}\)
\(g_{11}-g_{7}\)
\(-g_{10}+g_{9}\)
\(4g_{-15}\)
\(-2g_{-26}+2g_{-28}\)
\(2g_{-27}\)
\(g_{16}+g_{12}\)
\(4g_{-23}\)
\(2g_{-22}-2g_{-25}\)
\(-4g_{-19}\)
\(g_{19}\)
\(-g_{25}+g_{22}\)
\(g_{23}\)
\(-g_{-12}-g_{-16}\)
\(-2g_{27}\)
\(-g_{28}+g_{26}\)
\(-g_{15}\)
\(g_{-9}-g_{-10}\)
\(-g_{-7}+g_{-11}\)
\(-2g_{30}\)
\(g_{21}-g_{18}\)
\(-2g_{-35}\)
\(g_{-3}+g_{-5}\)
\(-g_{-17}-g_{-20}\)
\(2g_{24}\)
\(2g_{-34}\)
\(g_{-13}-g_{-14}\)
\(-2g_{-36}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}+\omega_{3}-\omega_{4}\)
\(-\omega_{2}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)		\(\omega_{3}+\omega_{4}\)
\(-\omega_{3}+2\omega_{4}\)
\(2\omega_{3}-\omega_{4}\)
\(0\)
\(0\)
\(-2\omega_{3}+\omega_{4}\)
\(\omega_{3}-2\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(2\omega_{1}+\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(2\omega_{1}-\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(2\omega_{1}-\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-2\omega_{2}+\omega_{3}\)
\(-\omega_{1}-\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-2\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)
\(-2\omega_{2}-\omega_{4}\)		\(2\omega_{2}+\omega_{4}\)
\(\omega_{1}+\omega_{4}\)
\(2\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{4}\)
\(\omega_{1}+\omega_{3}-\omega_{4}\)
\(2\omega_{2}-\omega_{3}\)
\(-\omega_{2}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-2\omega_{1}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(2\omega_{1}-2\omega_{2}-\omega_{3}\)
\(-2\omega_{1}+\omega_{3}-\omega_{4}\)
\(-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}+\omega_{3}-\omega_{4}\)
\(-\omega_{2}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)		\(\omega_{3}+\omega_{4}\)
\(-\omega_{3}+2\omega_{4}\)
\(2\omega_{3}-\omega_{4}\)
\(0\)
\(0\)
\(-2\omega_{3}+\omega_{4}\)
\(\omega_{3}-2\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(2\omega_{1}+\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(2\omega_{1}-\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(2\omega_{1}-\omega_{4}\)
\(-\omega_{1}+\omega_{3}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-2\omega_{2}+\omega_{3}\)
\(-\omega_{1}-\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-2\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)
\(-2\omega_{2}-\omega_{4}\)		\(2\omega_{2}+\omega_{4}\)
\(\omega_{1}+\omega_{4}\)
\(2\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{4}\)
\(\omega_{1}+\omega_{3}-\omega_{4}\)
\(2\omega_{2}-\omega_{3}\)
\(-\omega_{2}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-2\omega_{1}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(2\omega_{1}-2\omega_{2}-\omega_{3}\)
\(-2\omega_{1}+\omega_{3}-\omega_{4}\)
\(-\omega_{2}-\omega_{3}\)
\(-2\omega_{1}-\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}
\oplus M_{-\omega_{1}-\omega_{4}}\)		\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{-\omega_{3}+2\omega_{4}}\oplus M_{2\omega_{3}-\omega_{4}}\oplus 2M_{0}\oplus M_{-2\omega_{3}+\omega_{4}}
\oplus M_{\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{1}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-2\omega_{1}+2\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{4}}
\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{-2\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{-2\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}\oplus M_{-2\omega_{2}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}+\omega_{4}}\oplus M_{2\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}
\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{4}}\oplus M_{2\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{2\omega_{1}-2\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{3}-\omega_{4}}
\oplus M_{-2\omega_{1}-\omega_{3}}\)
Isotypic character\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}
\oplus M_{-\omega_{1}-\omega_{4}}\)		\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{-\omega_{3}+2\omega_{4}}\oplus M_{2\omega_{3}-\omega_{4}}\oplus 2M_{0}\oplus M_{-2\omega_{3}+\omega_{4}}
\oplus M_{\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{1}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-2\omega_{1}+2\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{4}}
\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{-2\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{-2\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}\oplus M_{-2\omega_{2}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}+\omega_{4}}\oplus M_{2\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}
\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{4}}\oplus M_{2\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{2\omega_{1}-2\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{3}-\omega_{4}}
\oplus M_{-2\omega_{1}-\omega_{3}}\)

Semisimple subalgebra: W_{1}+W_{5}
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 721 arithmetic operations while solving the Serre relations polynomial system.