Subalgebra \(C^{1}_3+A^{1}_1\) ↪ \(E^{1}_6\)
104 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle C^{1}_3+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) .
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: \(\displaystyle C^{1}_3\): (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, -1, -1, 0): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \(\displaystyle g_{-32}+g_{-33}\), \(\displaystyle g_{6}+g_{1}\), \(\displaystyle g_{15}\), \(\displaystyle g_{-4}\)
Positive simple generators: \(\displaystyle g_{33}+g_{32}\), \(\displaystyle g_{-1}+g_{-6}\), \(\displaystyle g_{-15}\), \(\displaystyle g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}}\)
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 5) ; the vectors are over the primal subalgebra.\(g_{14}+g_{13}\)\(g_{36}\)\(g_{28}+g_{26}\)\(g_{8}\)\(g_{4}\)
weight\(\omega_{2}\)\(2\omega_{1}\)\(\omega_{1}+\omega_{4}\)\(\omega_{3}+\omega_{4}\)\(2\omega_{4}\)
Isotypic module decomposition over primal subalgebra (total 5 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0, 0)\(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0, 0)\(\displaystyle V_{\omega_{1}+\omega_{4}} \) → (1, 0, 0, 1)\(\displaystyle V_{\omega_{3}+\omega_{4}} \) → (0, 0, 1, 1)\(\displaystyle V_{2\omega_{4}} \) → (0, 0, 0, 2)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
\(g_{14}+g_{13}\)
\(-g_{20}+g_{17}\)
\(g_{-18}+g_{-21}\)
\(-g_{31}-g_{29}\)
\(g_{-1}-g_{-6}\)
\(-g_{33}+g_{32}\)
\(-h_{6}+h_{1}\)
\(h_{5}-h_{3}\)
\(-g_{-32}+g_{-33}\)
\(2g_{6}-2g_{1}\)
\(-g_{-29}-g_{-31}\)
\(2g_{21}+2g_{18}\)
\(-g_{-17}+g_{-20}\)
\(g_{-13}+g_{-14}\)
Semisimple subalgebra component.
\(-g_{36}\)
\(g_{14}-g_{13}\)
\(2g_{-24}\)
\(-g_{20}-g_{17}\)
\(2g_{-18}-2g_{-21}\)
\(-g_{31}+g_{29}\)
\(-4g_{-15}\)
\(2g_{-1}+2g_{-6}\)
\(-g_{33}-g_{32}\)
\(-4h_{5}-4h_{4}-4h_{3}\)
\(2h_{6}+2h_{1}\)
\(2h_{6}+3h_{5}+4h_{4}+3h_{3}+2h_{2}+2h_{1}\)
\(-4g_{6}-4g_{1}\)
\(8g_{15}\)
\(2g_{-32}+2g_{-33}\)
\(4g_{-29}-4g_{-31}\)
\(-4g_{21}+4g_{18}\)
\(4g_{-17}+4g_{-20}\)
\(-8g_{24}\)
\(-4g_{-13}+4g_{-14}\)
\(8g_{-36}\)
\(g_{28}+g_{26}\)
\(g_{-7}+g_{-11}\)
\(g_{25}+g_{22}\)
\(-g_{-3}+g_{-5}\)
\(g_{-12}-g_{-16}\)
\(g_{10}+g_{9}\)
\(-g_{-9}-g_{-10}\)
\(-g_{16}+g_{12}\)
\(g_{5}-g_{3}\)
\(-g_{-22}-g_{-25}\)
\(-g_{11}-g_{7}\)
\(g_{-26}+g_{-28}\)
\(g_{8}\)
\(g_{23}\)
\(-g_{2}\)
\(-g_{28}+g_{26}\)
\(g_{19}\)
\(-g_{-7}+g_{-11}\)
\(-2g_{30}\)
\(-g_{25}+g_{22}\)
\(g_{-3}+g_{-5}\)
\(-g_{-12}-g_{-16}\)
\(2g_{35}\)
\(-2g_{27}\)
\(2g_{-34}\)
\(-g_{10}+g_{9}\)
\(g_{-9}-g_{-10}\)
\(2g_{34}\)
\(2g_{-27}\)
\(-2g_{-35}\)
\(g_{16}+g_{12}\)
\(-g_{5}-g_{3}\)
\(2g_{-22}-2g_{-25}\)
\(-2g_{-30}\)
\(g_{11}-g_{7}\)
\(-4g_{-19}\)
\(-2g_{-26}+2g_{-28}\)
\(-4g_{-2}\)
\(4g_{-23}\)
\(-4g_{-8}\)
Semisimple subalgebra component.
\(-g_{4}\)
\(h_{4}\)
\(2g_{-4}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}\)		\(2\omega_{1}\)
\(\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(2\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(2\omega_{1}-2\omega_{2}\)
\(-\omega_{2}\)
\(-2\omega_{1}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(\omega_{3}+\omega_{4}\)
\(2\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{1}+\omega_{4}\)
\(2\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(2\omega_{1}-\omega_{3}+\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}\)
\(-2\omega_{1}+\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(2\omega_{1}-\omega_{3}-\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-2\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-2\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}\)		\(2\omega_{1}\)
\(\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-2\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(2\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(2\omega_{1}-2\omega_{2}\)
\(-\omega_{2}\)
\(-2\omega_{1}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(\omega_{3}+\omega_{4}\)
\(2\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{1}+\omega_{4}\)
\(2\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{2}+\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(2\omega_{1}-\omega_{3}+\omega_{4}\)
\(2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}\)
\(-2\omega_{1}+\omega_{3}+\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(2\omega_{1}-\omega_{3}-\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(-2\omega_{1}+2\omega_{2}-\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{2}-\omega_{4}\)
\(-2\omega_{2}+\omega_{3}+\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-2\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus 2M_{0}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{2\omega_{1}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+2\omega_{2}}
\oplus 3M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{2\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}}\)		\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}-\omega_{4}}\)		\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}+\omega_{4}}\oplus M_{2\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{2\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-2\omega_{1}+\omega_{3}+\omega_{4}}
\oplus M_{-2\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{3}+\omega_{4}}\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}
\oplus M_{2\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{2\omega_{1}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-2\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}
\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)
Isotypic character\(\displaystyle M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus 2M_{0}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{2\omega_{1}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+2\omega_{2}}
\oplus 3M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{2\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-2\omega_{1}}\)		\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}
\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}-\omega_{4}}\)		\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}+\omega_{4}}\oplus M_{2\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{2\omega_{1}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-2\omega_{1}+\omega_{3}+\omega_{4}}
\oplus M_{-2\omega_{2}+\omega_{3}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{3}+\omega_{4}}\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{2\omega_{1}-2\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}
\oplus M_{2\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{2\omega_{1}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}-\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-2\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}
\oplus M_{-2\omega_{1}+2\omega_{2}-\omega_{3}-\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)		\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)

Semisimple subalgebra: W_{2}+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 715 arithmetic operations while solving the Serre relations polynomial system.