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

Subalgebra type: \(\displaystyle E^{1}_6\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle D^{1}_5\) .
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: \(\displaystyle E^{1}_6\): (1, 2, 2, 3, 2, 1): 2, (0, 0, -1, 0, 0, 0): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, 0, 0, 0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 78.
Negative simple generators: \(\displaystyle g_{-36}\), \(\displaystyle g_{3}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\), \(\displaystyle g_{6}\)
Positive simple generators: \(\displaystyle g_{36}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\), \(\displaystyle g_{-6}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}}\)
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 1) ; the vectors are over the primal subalgebra.\(g_{1}\)
weight\(\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 1 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0, 0, 0, 0)
Module label \(W_{1}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{1}\)
\(g_{7}\)
\(-g_{12}\)
\(-g_{17}\)
\(g_{18}\)
\(-g_{-31}\)
\(g_{22}\)
\(-g_{24}\)
\(g_{-28}\)
\(g_{26}\)
\(-g_{27}\)
\(-g_{-25}\)
\(g_{-23}\)
\(g_{29}\)
\(-g_{30}\)
\(g_{-20}\)
\(g_{-21}\)
\(-g_{-19}\)
\(-g_{32}\)
\(-g_{33}\)
\(-g_{-16}\)
\(g_{-14}\)
\(g_{-15}\)
\(-g_{-13}\)
\(-g_{34}\)
\(g_{-11}\)
\(-g_{-10}\)
\(g_{-8}\)
\(g_{-9}\)
\(-g_{35}\)
\(-g_{-6}\)
\(g_{-5}\)
\(-g_{-4}\)
\(g_{-2}\)
\(g_{-3}\)
\(-g_{36}\)
\(-h_{6}\)
\(h_{5}\)
\(-h_{4}\)
\(h_{2}\)
\(h_{3}\)
\(h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}\)
\(-g_{5}\)
\(2g_{6}\)
\(g_{4}\)
\(-g_{3}\)
\(-g_{2}\)
\(g_{-36}\)
\(-g_{10}\)
\(g_{11}\)
\(g_{9}\)
\(g_{8}\)
\(-g_{-35}\)
\(-g_{15}\)
\(-g_{14}\)
\(g_{16}\)
\(g_{13}\)
\(g_{-34}\)
\(-g_{19}\)
\(g_{21}\)
\(-g_{-32}\)
\(g_{20}\)
\(-g_{-33}\)
\(g_{-30}\)
\(-g_{23}\)
\(g_{25}\)
\(-g_{-29}\)
\(-g_{-27}\)
\(g_{-26}\)
\(g_{28}\)
\(g_{-24}\)
\(-g_{-22}\)
\(g_{31}\)
\(g_{-18}\)
\(-g_{-17}\)
\(g_{-12}\)
\(g_{-7}\)
\(g_{-1}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{2}\)
\(-\omega_{2}+\omega_{4}\)
\(\omega_{3}-\omega_{4}+\omega_{5}\)
\(\omega_{1}-\omega_{3}+\omega_{5}\)
\(\omega_{3}-\omega_{5}+\omega_{6}\)
\(-\omega_{1}+\omega_{5}\)
\(\omega_{1}-\omega_{3}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{3}-\omega_{6}\)
\(-\omega_{1}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{4}+\omega_{6}\)
\(\omega_{1}-\omega_{3}+\omega_{4}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}+\omega_{6}\)
\(-\omega_{1}+\omega_{4}-\omega_{6}\)
\(\omega_{1}-\omega_{2}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{6}\)
\(\omega_{2}-\omega_{3}+\omega_{6}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(\omega_{1}-\omega_{2}+\omega_{5}-\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{5}\)
\(-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{5}-\omega_{6}\)
\(\omega_{2}-\omega_{3}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{5}\)
\(\omega_{1}-\omega_{2}+\omega_{4}-\omega_{5}\)
\(-\omega_{4}+\omega_{5}+\omega_{6}\)
\(-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{4}-\omega_{5}\)
\(\omega_{2}-\omega_{3}+\omega_{4}-\omega_{5}\)
\(\omega_{1}+\omega_{3}-\omega_{4}\)
\(-\omega_{5}+2\omega_{6}\)
\(-\omega_{4}+2\omega_{5}-\omega_{6}\)
\(-\omega_{2}-\omega_{3}+2\omega_{4}-\omega_{5}\)
\(-\omega_{1}+2\omega_{3}-\omega_{4}\)
\(2\omega_{2}-\omega_{4}\)
\(2\omega_{1}-\omega_{3}\)
\(0\)
\(0\)
\(0\)
\(0\)
\(0\)
\(0\)
\(\omega_{4}-2\omega_{5}+\omega_{6}\)
\(\omega_{5}-2\omega_{6}\)
\(\omega_{2}+\omega_{3}-2\omega_{4}+\omega_{5}\)
\(-2\omega_{2}+\omega_{4}\)
\(\omega_{1}-2\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+\omega_{3}\)
\(\omega_{2}+\omega_{3}-\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{4}-\omega_{5}-\omega_{6}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}+\omega_{5}\)
\(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}+\omega_{5}\)
\(-\omega_{1}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{5}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{5}+\omega_{6}\)
\(\omega_{2}+\omega_{3}-\omega_{4}-\omega_{6}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{5}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}+\omega_{5}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(-\omega_{2}+\omega_{3}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}-\omega_{5}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{5}\)
\(-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{1}-\omega_{4}+\omega_{6}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}-\omega_{6}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{6}\)
\(\omega_{1}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(-\omega_{3}+\omega_{6}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(\omega_{1}-\omega_{5}\)
\(-\omega_{3}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}+\omega_{3}-\omega_{5}\)
\(-\omega_{3}+\omega_{4}-\omega_{5}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{2}\)
\(-\omega_{2}+\omega_{4}\)
\(\omega_{3}-\omega_{4}+\omega_{5}\)
\(\omega_{1}-\omega_{3}+\omega_{5}\)
\(\omega_{3}-\omega_{5}+\omega_{6}\)
\(-\omega_{1}+\omega_{5}\)
\(\omega_{1}-\omega_{3}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{3}-\omega_{6}\)
\(-\omega_{1}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{4}+\omega_{6}\)
\(\omega_{1}-\omega_{3}+\omega_{4}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}+\omega_{6}\)
\(-\omega_{1}+\omega_{4}-\omega_{6}\)
\(\omega_{1}-\omega_{2}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{6}\)
\(\omega_{2}-\omega_{3}+\omega_{6}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(\omega_{1}-\omega_{2}+\omega_{5}-\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{5}\)
\(-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{5}-\omega_{6}\)
\(\omega_{2}-\omega_{3}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{5}\)
\(\omega_{1}-\omega_{2}+\omega_{4}-\omega_{5}\)
\(-\omega_{4}+\omega_{5}+\omega_{6}\)
\(-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{4}-\omega_{5}\)
\(\omega_{2}-\omega_{3}+\omega_{4}-\omega_{5}\)
\(\omega_{1}+\omega_{3}-\omega_{4}\)
\(-\omega_{5}+2\omega_{6}\)
\(-\omega_{4}+2\omega_{5}-\omega_{6}\)
\(-\omega_{2}-\omega_{3}+2\omega_{4}-\omega_{5}\)
\(-\omega_{1}+2\omega_{3}-\omega_{4}\)
\(2\omega_{2}-\omega_{4}\)
\(2\omega_{1}-\omega_{3}\)
\(0\)
\(0\)
\(0\)
\(0\)
\(0\)
\(0\)
\(\omega_{4}-2\omega_{5}+\omega_{6}\)
\(\omega_{5}-2\omega_{6}\)
\(\omega_{2}+\omega_{3}-2\omega_{4}+\omega_{5}\)
\(-2\omega_{2}+\omega_{4}\)
\(\omega_{1}-2\omega_{3}+\omega_{4}\)
\(-2\omega_{1}+\omega_{3}\)
\(\omega_{2}+\omega_{3}-\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{4}-\omega_{5}-\omega_{6}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}+\omega_{5}\)
\(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}+\omega_{5}\)
\(-\omega_{1}-\omega_{3}+\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{5}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{5}+\omega_{6}\)
\(\omega_{2}+\omega_{3}-\omega_{4}-\omega_{6}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{5}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}+\omega_{5}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(-\omega_{2}+\omega_{3}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}-\omega_{5}+\omega_{6}\)
\(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{5}\)
\(-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{5}+\omega_{6}\)
\(\omega_{1}-\omega_{4}+\omega_{6}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}-\omega_{6}\)
\(-\omega_{1}+\omega_{2}-\omega_{6}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{6}\)
\(-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{6}\)
\(\omega_{1}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(-\omega_{3}+\omega_{6}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{5}-\omega_{6}\)
\(\omega_{1}-\omega_{5}\)
\(-\omega_{3}+\omega_{5}-\omega_{6}\)
\(-\omega_{1}+\omega_{3}-\omega_{5}\)
\(-\omega_{3}+\omega_{4}-\omega_{5}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}+\omega_{2}-\omega_{4}+\omega_{6}}\oplus M_{-\omega_{5}+2\omega_{6}}\oplus M_{-\omega_{4}+\omega_{5}+\omega_{6}}
\oplus M_{\omega_{1}-\omega_{2}+\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_{4}+\omega_{6}}\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}-\omega_{5}+\omega_{6}}\oplus M_{\omega_{3}-\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_{5}+\omega_{6}}
\oplus M_{\omega_{1}-\omega_{3}+\omega_{5}}\oplus M_{\omega_{3}-\omega_{4}+\omega_{5}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}+\omega_{5}}
\oplus M_{\omega_{2}+\omega_{3}-2\omega_{4}+\omega_{5}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{3}-\omega_{4}}
\oplus M_{2\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{5}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{4}+\omega_{5}-\omega_{6}}
\oplus M_{-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{6}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{6}}\oplus M_{-\omega_{3}+\omega_{6}}
\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{6}}\oplus M_{-\omega_{1}+\omega_{4}-\omega_{5}+\omega_{6}}\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_{2}-\omega_{5}+\omega_{6}}\oplus M_{\omega_{4}-2\omega_{5}+\omega_{6}}
\oplus M_{-\omega_{1}+\omega_{5}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{5}}\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}}\oplus M_{\omega_{1}-2\omega_{3}+\omega_{4}}
\oplus 6M_{0}\oplus M_{-\omega_{1}+2\omega_{3}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{4}}\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_{3}-\omega_{5}}\oplus M_{\omega_{1}-\omega_{5}}
\oplus M_{-\omega_{4}+2\omega_{5}-\omega_{6}}\oplus M_{\omega_{1}-\omega_{2}+\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_{4}+\omega_{5}-\omega_{6}}
\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}-\omega_{6}}\oplus M_{\omega_{3}-\omega_{6}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}-\omega_{6}}
\oplus M_{\omega_{2}+\omega_{3}-\omega_{4}-\omega_{6}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{5}+\omega_{6}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{5}}\oplus M_{-2\omega_{2}+\omega_{4}}\oplus M_{-\omega_{1}-\omega_{3}+\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{2}-\omega_{3}+2\omega_{4}-\omega_{5}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{4}-\omega_{5}}
\oplus M_{-\omega_{3}+\omega_{4}-\omega_{5}}\oplus M_{-\omega_{1}+\omega_{3}-\omega_{5}}\oplus M_{-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{5}-\omega_{6}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{5}-\omega_{6}}\oplus M_{-\omega_{3}+\omega_{5}-\omega_{6}}\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{5}-\omega_{6}}
\oplus M_{-\omega_{1}+\omega_{4}-\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_{2}-\omega_{6}}\oplus M_{\omega_{4}-\omega_{5}-\omega_{6}}\oplus M_{\omega_{5}-2\omega_{6}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{6}}\)
Isotypic character\(\displaystyle M_{\omega_{1}+\omega_{2}-\omega_{4}+\omega_{6}}\oplus M_{-\omega_{5}+2\omega_{6}}\oplus M_{-\omega_{4}+\omega_{5}+\omega_{6}}
\oplus M_{\omega_{1}-\omega_{2}+\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_{4}+\omega_{6}}\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}-\omega_{5}+\omega_{6}}\oplus M_{\omega_{3}-\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_{5}+\omega_{6}}
\oplus M_{\omega_{1}-\omega_{3}+\omega_{5}}\oplus M_{\omega_{3}-\omega_{4}+\omega_{5}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}+\omega_{5}}
\oplus M_{\omega_{2}+\omega_{3}-2\omega_{4}+\omega_{5}}\oplus M_{\omega_{2}}\oplus M_{2\omega_{1}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{3}-\omega_{4}}
\oplus M_{2\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{5}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{4}+\omega_{5}-\omega_{6}}
\oplus M_{-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{6}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{6}}\oplus M_{-\omega_{3}+\omega_{6}}
\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{6}}\oplus M_{-\omega_{1}+\omega_{4}-\omega_{5}+\omega_{6}}\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_{2}-\omega_{5}+\omega_{6}}\oplus M_{\omega_{4}-2\omega_{5}+\omega_{6}}
\oplus M_{-\omega_{1}+\omega_{5}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{5}}\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}}\oplus M_{\omega_{1}-2\omega_{3}+\omega_{4}}
\oplus 6M_{0}\oplus M_{-\omega_{1}+2\omega_{3}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{4}}\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_{3}-\omega_{5}}\oplus M_{\omega_{1}-\omega_{5}}
\oplus M_{-\omega_{4}+2\omega_{5}-\omega_{6}}\oplus M_{\omega_{1}-\omega_{2}+\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_{4}+\omega_{5}-\omega_{6}}
\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}-\omega_{6}}\oplus M_{\omega_{3}-\omega_{6}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}-\omega_{6}}
\oplus M_{\omega_{2}+\omega_{3}-\omega_{4}-\omega_{6}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{5}+\omega_{6}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{5}}\oplus M_{-2\omega_{2}+\omega_{4}}\oplus M_{-\omega_{1}-\omega_{3}+\omega_{4}}
\oplus M_{-2\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{2}-\omega_{3}+2\omega_{4}-\omega_{5}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{4}-\omega_{5}}
\oplus M_{-\omega_{3}+\omega_{4}-\omega_{5}}\oplus M_{-\omega_{1}+\omega_{3}-\omega_{5}}\oplus M_{-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{5}-\omega_{6}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}+\omega_{5}-\omega_{6}}\oplus M_{-\omega_{3}+\omega_{5}-\omega_{6}}\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}+\omega_{5}-\omega_{6}}
\oplus M_{-\omega_{1}+\omega_{4}-\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_{2}-\omega_{6}}\oplus M_{\omega_{4}-\omega_{5}-\omega_{6}}\oplus M_{\omega_{5}-2\omega_{6}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{4}-\omega_{6}}\)

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