Subalgebra \(3A^{1}_2\) ↪ \(E^{1}_6\)
117 out of 119
Computations done by the calculator project.

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

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 701 arithmetic operations while solving the Serre relations polynomial system.