Subalgebra \(2A^{1}_2+A^{4}_1\) ↪ \(E^{1}_6\)
112 out of 119
Computations done by the calculator project.

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

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 3798 arithmetic operations while solving the Serre relations polynomial system.