Up to linear equivalence, there are total 59 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents. The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra.
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 3, 4, 2): 2 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-24}\) Positive simple generators: \(\displaystyle g_{24}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 14V_{\omega_{1}}\oplus 21V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}}\oplus V_{2\psi_{3}}\oplus V_{\omega_{1}+\psi_{3}} \oplus V_{2\psi_{1}-\psi_{2}+\psi_{3}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+2\psi_{1}-\psi_{2}}\oplus V_{4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{3}}\oplus V_{-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-\psi_{2}+\psi_{3}}\oplus V_{2\psi_{1}-2\psi_{2}+\psi_{3}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\psi_{2}-\psi_{3}} \oplus V_{2\psi_{1}-\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{3}}\oplus V_{-2\psi_{2}+2\psi_{3}}\oplus V_{-2\psi_{1}+\psi_{2}+\psi_{3}} \oplus V_{\omega_{1}-2\psi_{1}+\psi_{2}}\oplus 3V_{0}\oplus V_{\omega_{1}-\psi_{3}}\oplus V_{2\psi_{1}-\psi_{2}-\psi_{3}}\oplus V_{2\psi_{2}-2\psi_{3}} \oplus V_{-2\psi_{1}+\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{-\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}-\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{1}+\psi_{2}-\psi_{3}}\oplus V_{-2\psi_{3}}\) Made total 278 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1\) ↪ \(F^{1}_4\) 2 out of 59
Subalgebra type: \(\displaystyle A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Centralizer: \(\displaystyle A^{1}_3\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_1\) Basis of Cartan of centralizer: 3 vectors:
(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+A^{1}_1\) , \(\displaystyle 2A^{2}_1\) , \(\displaystyle A^{4}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+A^{2}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle 3A^{2}_1\) , \(\displaystyle A^{1}_2+A^{2}_1\) , \(\displaystyle B^{1}_2+A^{2}_1\) , \(\displaystyle A^{1}_3+A^{2}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 6, 4): 4 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-21}\) Positive simple generators: \(\displaystyle g_{21}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 7V_{2\omega_{1}}\oplus 8V_{\omega_{1}}\oplus 15V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+\psi_{3}}\oplus V_{2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{2}-\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-\psi_{3}} \oplus V_{2\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}+2\psi_{3}}\oplus V_{2\psi_{1}-4\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}-2\psi_{1}+\psi_{3}}\oplus V_{\omega_{1}-2\psi_{2}+\psi_{3}}\oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-\psi_{3}} \oplus V_{\omega_{1}-\psi_{3}}\oplus V_{-2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{2\psi_{1}-2\psi_{3}}\oplus V_{-2\psi_{1}-2\psi_{2}+2\psi_{3}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}-2\psi_{3}}\) Made total 177381398 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1\) ↪ \(F^{1}_4\) 3 out of 59
Subalgebra type: \(\displaystyle A^{3}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{8}_1+A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{3}_1\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0), (0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1\) , \(\displaystyle A^{9}_1+A^{3}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (3, 6, 8, 4): 6 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-14}+g_{-21}\) Positive simple generators: \(\displaystyle g_{21}+g_{14}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{3\omega_{1}}\oplus 6V_{2\omega_{1}}\oplus 10V_{\omega_{1}}\oplus 6V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+2\psi_{1}}\oplus V_{2\omega_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}+\psi_{2}} \oplus V_{4\psi_{1}}\oplus V_{2\omega_{1}+\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+2\psi_{1}-\psi_{2}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{\psi_{2}}\oplus V_{3\omega_{1}-2\psi_{1}}\oplus V_{2\omega_{1}-\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}-2\psi_{1}+\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{-\psi_{2}} \oplus V_{\omega_{1}-2\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{1}}\) Made total 11725162 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1\) ↪ \(F^{1}_4\) 4 out of 59
Subalgebra type: \(\displaystyle A^{4}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{2}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0), (0, 1, 0, 1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{4}_1+A^{2}_1\) , \(\displaystyle A^{8}_1+A^{4}_1\) , \(\displaystyle A^{2}_2+A^{4}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (4, 6, 8, 4): 8 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-14}+g_{-18}\) Positive simple generators: \(\displaystyle 2g_{18}+2g_{14}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}}\oplus 13V_{2\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}-\psi_{1}+2\psi_{2}} \oplus V_{2\omega_{1}+\psi_{2}}\oplus V_{-\psi_{1}+3\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}+\psi_{2}} \oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+3\psi_{2}} \oplus V_{\psi_{1}}\oplus V_{2\omega_{1}-\psi_{2}}\oplus V_{2\omega_{1}+\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+2\psi_{1}-4\psi_{2}}\oplus V_{-\psi_{1}}\oplus V_{2\psi_{1}-3\psi_{2}}\oplus V_{\psi_{1}-3\psi_{2}}\) Made total 256983082 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{6}_1\) ↪ \(F^{1}_4\) 5 out of 59
Subalgebra type: \(\displaystyle A^{6}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{6}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{6}_1\) Basis of Cartan of centralizer: 1 vectors:
(1, 1, 0, -1) Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{6}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{6}_1\): (4, 8, 12, 6): 12 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-9}+g_{-15}+g_{-20}\) Positive simple generators: \(\displaystyle 2g_{20}+g_{15}+2g_{9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}12\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus 6V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{4\omega_{1}+2\psi}\oplus V_{\omega_{1}+3\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{3\omega_{1}+\psi} \oplus V_{4\omega_{1}}\oplus V_{2\psi}\oplus V_{\omega_{1}+\psi}\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}-\psi}\oplus V_{4\omega_{1}-2\psi}\oplus V_{0} \oplus V_{\omega_{1}-\psi}\oplus V_{2\omega_{1}-2\psi}\oplus V_{-2\psi}\oplus V_{\omega_{1}-3\psi}\oplus V_{2\omega_{1}-4\psi}\) Made total 3617844 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1\) ↪ \(F^{1}_4\) 6 out of 59
Subalgebra type: \(\displaystyle A^{8}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle G^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{8}_1\) Basis of Cartan of centralizer: 2 vectors:
(1, 1, 0, 0), (1, 0, -2, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1\) , \(\displaystyle A^{8}_1+A^{4}_1\) , \(\displaystyle A^{28}_1+A^{8}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{1}_2+A^{8}_1\) , \(\displaystyle G^{1}_2+A^{8}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 12, 8): 16 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-10}+g_{-15}\) Positive simple generators: \(\displaystyle 2g_{15}+2g_{10}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 7V_{4\omega_{1}}\oplus V_{2\omega_{1}}\oplus 14V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{1}+6\psi_{2}}\oplus V_{4\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{1}+6\psi_{2}}\oplus V_{4\omega_{1}+2\psi_{1}+2\psi_{2}} \oplus V_{2\psi_{1}+4\psi_{2}}\oplus V_{4\omega_{1}+2\psi_{2}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\psi_{2}}\oplus V_{2\psi_{1}} \oplus V_{2\omega_{1}}\oplus V_{4\omega_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{4\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}} \oplus V_{4\omega_{1}-2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-6\psi_{2}} \oplus V_{-4\psi_{1}-6\psi_{2}}\) Made total 5692828 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1\) ↪ \(F^{1}_4\) 7 out of 59
Subalgebra type: \(\displaystyle A^{9}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{3}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{8}_1+A^{1}_1\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -2, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{9}_1+A^{3}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (5, 10, 14, 8): 18 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-10}+g_{-14}+g_{-15}\) Positive simple generators: \(\displaystyle 2g_{15}+g_{14}+2g_{10}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{5\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus 2V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{\omega_{1}+6\psi}\oplus V_{5\omega_{1}+2\psi}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}} \oplus V_{\omega_{1}+2\psi}\oplus V_{5\omega_{1}-2\psi}\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi} \oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\oplus V_{\omega_{1}-6\psi}\) Made total 476289 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1\) ↪ \(F^{1}_4\) 8 out of 59
Subalgebra type: \(\displaystyle A^{10}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle 2A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0), (0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 10, 14, 8): 20 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-8}+g_{-16}\) Positive simple generators: \(\displaystyle 4g_{16}+3g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 4V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus V_{2\omega_{1}}\oplus 6V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\psi_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\psi_{2}}\oplus V_{3\omega_{1}+2\psi_{1}-\psi_{2}} \oplus V_{4\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{3\omega_{1}-2\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}}\oplus V_{4\omega_{1}-2\psi_{1}} \oplus V_{3\omega_{1}-\psi_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{2}}\) Made total 16128908 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1\) ↪ \(F^{1}_4\) 9 out of 59
Subalgebra type: \(\displaystyle A^{11}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{11}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 11, 16, 8): 22 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-8}+g_{-9}+g_{-16}\) Positive simple generators: \(\displaystyle 4g_{16}+g_{9}+3g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus 3V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{5\omega_{1}+2\psi}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}}\oplus V_{5\omega_{1}-2\psi} \oplus 3V_{2\omega_{1}}\oplus 2V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\) Made total 843481 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{12}_1\) ↪ \(F^{1}_4\) 10 out of 59
Subalgebra type: \(\displaystyle A^{12}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (6, 12, 16, 8): 24 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-2}-2g_{-9}+g_{-11}+g_{-18}\) Positive simple generators: \(\displaystyle 4g_{18}+g_{16}+2g_{11}-g_{9}+4g_{5}+4g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/6\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}24\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{6\omega_{1}}\oplus 4V_{4\omega_{1}}\oplus 6V_{2\omega_{1}}\) Made total 12399874 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{28}_1\) ↪ \(F^{1}_4\) 11 out of 59
Subalgebra type: \(\displaystyle A^{28}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{8}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{28}_1+A^{8}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (10, 18, 24, 12): 56 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-13}\) Positive simple generators: \(\displaystyle 6g_{13}+6g_{2}+10g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 5V_{6\omega_{1}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\psi}\oplus V_{6\omega_{1}+\psi}\oplus V_{6\omega_{1}}\oplus V_{6\omega_{1}-\psi}\oplus V_{6\omega_{1}-2\psi} \oplus V_{2\omega_{1}}\oplus V_{\psi}\oplus V_{0}\oplus V_{-\psi}\) Made total 7149101 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1\) ↪ \(F^{1}_4\) 12 out of 59
Subalgebra type: \(\displaystyle A^{35}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{35}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 19, 28, 16): 70 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-4}+g_{-8}+g_{-9}\) Positive simple generators: \(\displaystyle 9g_{9}+5g_{8}+8g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 2V_{9\omega_{1}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{9\omega_{1}+2\psi}\oplus V_{10\omega_{1}}\oplus V_{9\omega_{1}-2\psi}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi} \oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\) Made total 7681 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{36}_1\) ↪ \(F^{1}_4\) 13 out of 59
Subalgebra type: \(\displaystyle A^{36}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (10, 20, 28, 16): 72 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-4}+g_{-5}+g_{-6}+g_{-11}\) Positive simple generators: \(\displaystyle 9g_{11}+5g_{6}+g_{5}+8g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/18\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}72\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\) Made total 4345721 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{60}_1\) ↪ \(F^{1}_4\) 14 out of 59
Subalgebra type: \(\displaystyle A^{60}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{60}_1\): (14, 26, 36, 20): 120 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-7}+g_{-9}\) Positive simple generators: \(\displaystyle 8g_{9}+10g_{7}+18g_{2}+14g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/30\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}120\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus 2V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\) Made total 1189835 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{156}_1\) ↪ \(F^{1}_4\) 15 out of 59
Subalgebra type: \(\displaystyle A^{156}_1\) (click on type for detailed printout).
Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{156}_1\): (22, 42, 60, 32): 312 Dimension of subalgebra generated by predefined or computed generators: 3. Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}\) Positive simple generators: \(\displaystyle 16g_{4}+30g_{3}+42g_{2}+22g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/78\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}312\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{22\omega_{1}}\oplus V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{2\omega_{1}}\) Made total 21930 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_1\) ↪ \(F^{1}_4\) 16 out of 59
Subalgebra type: \(\displaystyle 2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) . Centralizer: \(\displaystyle B^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{1}_1\) Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0), (0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle 3A^{1}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle 4A^{1}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 3, 4, 2): 2, \(\displaystyle A^{1}_1\): (0, 1, 2, 2): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-16}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{16}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 5V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 10V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\psi_{2}} \oplus V_{\omega_{1}+\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}+2\psi_{1}-\psi_{2}} \oplus V_{\omega_{1}+2\psi_{1}-\psi_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}} \oplus V_{-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}} \oplus V_{\omega_{2}-\psi_{2}}\oplus V_{\omega_{1}-\psi_{2}}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{1}} \oplus V_{-2\psi_{2}}\) Made total 365 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+A^{1}_1\) ↪ \(F^{1}_4\) 17 out of 59
Subalgebra type: \(\displaystyle A^{2}_1+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0), (0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 6, 4): 4, \(\displaystyle A^{1}_1\): (1, 2, 2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{-14}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{14}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}} \oplus 4V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}}\oplus V_{2\omega_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}+\psi_{2}} \oplus V_{4\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+2\psi_{1}-\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{2}} \oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{1}}\) Made total 601440 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1\) ↪ \(F^{1}_4\) 18 out of 59
Subalgebra type: \(\displaystyle 2A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle A^{2}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle 3A^{2}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 6, 4): 4, \(\displaystyle A^{2}_1\): (2, 2, 2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-20}+g_{-22}\), \(\displaystyle g_{-5}+g_{-11}\) Positive simple generators: \(\displaystyle g_{22}+g_{20}\), \(\displaystyle g_{11}+g_{5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 4V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 4V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{2}+2\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}+\psi}\oplus V_{2\psi} \oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{1}+\omega_{2}-\psi}\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}\oplus V_{2\omega_{1}-2\psi} \oplus V_{-2\psi}\) Made total 1845989 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{1}_1\) ↪ \(F^{1}_4\) 19 out of 59
Subalgebra type: \(\displaystyle A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle A^{8}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (3, 6, 8, 4): 6, \(\displaystyle A^{1}_1\): (1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-14}+g_{-21}\), \(\displaystyle g_{-1}\) Positive simple generators: \(\displaystyle g_{21}+g_{14}\), \(\displaystyle g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 5V_{\omega_{1}+\omega_{2}}\oplus 6V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+2\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+\psi} \oplus V_{2\omega_{1}+\psi}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\psi}\oplus V_{\omega_{1}+\omega_{2}-\psi} \oplus V_{2\omega_{1}-\psi}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{1}-2\psi}\oplus V_{-\psi}\) Made total 42926 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1+A^{2}_1\) ↪ \(F^{1}_4\) 20 out of 59
Subalgebra type: \(\displaystyle A^{4}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (4, 6, 8, 4): 8, \(\displaystyle A^{2}_1\): (0, 2, 4, 2): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-14}+g_{-18}\), \(\displaystyle g_{-13}\) Positive simple generators: \(\displaystyle 2g_{18}+2g_{14}\), \(\displaystyle g_{13}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 3V_{2\omega_{1}} \oplus 2V_{\omega_{2}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi}\oplus V_{\omega_{2}+3\psi}\oplus V_{2\omega_{1}+\omega_{2}+\psi} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi} \oplus V_{0}\oplus V_{\omega_{2}-3\psi}\oplus V_{2\omega_{1}-4\psi}\) Made total 287932 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{6}_1\) ↪ \(F^{1}_4\) 21 out of 59
Subalgebra type: \(\displaystyle 2A^{6}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{6}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{6}_1\): (4, 8, 12, 6): 12, \(\displaystyle A^{6}_1\): (2, 2, 0, 2): 12 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-14}+g_{-15}+g_{-16}\), \(\displaystyle g_{-1}-1/2g_{-2}-1/2g_{-4}\) Positive simple generators: \(\displaystyle 2g_{16}+g_{15}+2g_{14}\), \(\displaystyle -2g_{4}-4g_{2}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/3 & 0\\
0 & 1/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}12 & 0\\
0 & 12\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 3456369 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{1}_1\) ↪ \(F^{1}_4\) 22 out of 59
Subalgebra type: \(\displaystyle A^{8}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) . Centralizer: \(\displaystyle A^{3}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{8}_1+A^{1}_1\) Basis of Cartan of centralizer: 1 vectors:
(1, 0, -2, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 12, 8): 16, \(\displaystyle A^{1}_1\): (1, 2, 2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-10}+g_{-15}\), \(\displaystyle g_{-14}\) Positive simple generators: \(\displaystyle 2g_{15}+2g_{10}\), \(\displaystyle g_{14}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{4\omega_{1}+\omega_{2}}\oplus 3V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{2}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{\omega_{2}+6\psi}\oplus V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}}\oplus V_{\omega_{2}+2\psi} \oplus V_{4\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus V_{\omega_{2}-2\psi} \oplus V_{-4\psi}\oplus V_{\omega_{2}-6\psi}\) Made total 618 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1\) ↪ \(F^{1}_4\) 23 out of 59
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 12, 8): 16, \(\displaystyle A^{3}_1\): (2, 3, 4, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-12}+g_{-13}\), \(\displaystyle g_{-8}-g_{-9}\) Positive simple generators: \(\displaystyle 2g_{13}+2g_{12}\), \(\displaystyle -g_{9}+g_{8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\
0 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\
0 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{4\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}+\omega_{2}-2\psi} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{2}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\) Made total 499259 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{4}_1\) ↪ \(F^{1}_4\) 24 out of 59
Subalgebra type: \(\displaystyle A^{8}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 12, 8): 16, \(\displaystyle A^{4}_1\): (2, 4, 4, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-10}+g_{-15}\), \(\displaystyle g_{-5}-g_{-6}+g_{-11}\) Positive simple generators: \(\displaystyle 2g_{15}+2g_{10}\), \(\displaystyle g_{11}-g_{6}+g_{5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\
0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\
0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{4\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 2879078 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1+A^{3}_1\) ↪ \(F^{1}_4\) 25 out of 59
Subalgebra type: \(\displaystyle A^{9}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (5, 10, 14, 8): 18, \(\displaystyle A^{3}_1\): (1, 0, 2, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-12}+g_{-13}+g_{-14}\), \(\displaystyle g_{-1}-g_{-3}\) Positive simple generators: \(\displaystyle g_{14}+2g_{13}+2g_{12}\), \(\displaystyle -g_{3}+g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\
0 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\
0 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}} \oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\) Made total 820990 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{1}_1\) ↪ \(F^{1}_4\) 26 out of 59
Subalgebra type: \(\displaystyle A^{10}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 10, 14, 8): 20, \(\displaystyle A^{1}_1\): (0, 1, 2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-8}+g_{-16}\), \(\displaystyle g_{-9}\) Positive simple generators: \(\displaystyle 4g_{16}+3g_{8}\), \(\displaystyle g_{9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{4\omega_{1}+\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{3\omega_{1}+\omega_{2}} \oplus V_{4\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\) Made total 8127 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{2}_1\) ↪ \(F^{1}_4\) 27 out of 59
Subalgebra type: \(\displaystyle A^{10}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 10, 14, 8): 20, \(\displaystyle A^{2}_1\): (0, 2, 2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-5}+g_{-11}+g_{-16}\), \(\displaystyle -g_{-2}+g_{-9}\) Positive simple generators: \(\displaystyle 4g_{16}+3g_{11}+3g_{5}\), \(\displaystyle g_{9}-g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\
0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\
0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 58015 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1+A^{1}_1\) ↪ \(F^{1}_4\) 28 out of 59
Subalgebra type: \(\displaystyle A^{11}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 11, 16, 8): 22, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-8}+g_{-9}+g_{-16}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle 4g_{16}+g_{9}+3g_{8}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\) Made total 701 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{28}_1+A^{8}_1\) ↪ \(F^{1}_4\) 29 out of 59
Subalgebra type: \(\displaystyle A^{28}_1+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{28}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (10, 18, 24, 12): 56, \(\displaystyle A^{8}_1\): (0, 0, 4, 4): 16 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-1}+g_{-9}+g_{-10}\), \(\displaystyle g_{-3}+g_{-4}\) Positive simple generators: \(\displaystyle 6g_{10}+6g_{9}+10g_{1}\), \(\displaystyle 2g_{4}+2g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14 & 0\\
0 & 1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56 & 0\\
0 & 16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}+4\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 682784 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1+A^{1}_1\) ↪ \(F^{1}_4\) 30 out of 59
Subalgebra type: \(\displaystyle A^{35}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{35}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 19, 28, 16): 70, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 6. Negative simple generators: \(\displaystyle g_{-4}+g_{-8}+g_{-9}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle 9g_{9}+5g_{8}+8g_{4}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35 & 0\\
0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70 & 0\\
0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{9\omega_{1}+\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 541 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2\) ↪ \(F^{1}_4\) 31 out of 59
Subalgebra type: \(\displaystyle A^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) . Centralizer: \(\displaystyle A^{2}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0), (0, 0, 0, 1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_2+A^{2}_1\) , \(\displaystyle A^{1}_2+A^{8}_1\) , \(\displaystyle A^{2}_2+A^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}}\oplus 6V_{\omega_{2}}\oplus 6V_{\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}}\oplus V_{\psi_{1}+\psi_{2}}\oplus V_{\omega_{1}+\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}} \oplus V_{\omega_{1}+\omega_{2}}\oplus V_{-\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}-\psi_{1}+\psi_{2}} \oplus V_{2\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}-\psi_{1}} \oplus V_{\omega_{2}-\psi_{2}}\oplus V_{-2\psi_{1}+\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}}\oplus V_{\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{2}} \oplus V_{-\psi_{1}-\psi_{2}}\) Made total 361 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2\) ↪ \(F^{1}_4\) 32 out of 59
Subalgebra type: \(\displaystyle B^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) . Centralizer: \(\displaystyle 2A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\) Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0), (0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 3, 4, 2): 2, (-2, -2, -2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 10. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{8}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-8}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2\\
-2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 6V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{2}+\psi_{2}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{2}+2\psi_{1}-\psi_{2}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{2}-\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{-2\psi_{2}}\) Made total 11689686 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{1}_2\) ↪ \(F^{1}_4\) 33 out of 59
Subalgebra type: \(\displaystyle G^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: \(\displaystyle A^{8}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, 0) Contained up to conjugation as a direct summand of: \(\displaystyle G^{1}_2+A^{8}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (3, 6, 8, 4): 6, (-1, -3, -4, -2): 2 Dimension of subalgebra generated by predefined or computed generators: 14. Negative simple generators: \(\displaystyle g_{-14}+g_{-21}\), \(\displaystyle g_{23}\) Positive simple generators: \(\displaystyle g_{21}+g_{14}\), \(\displaystyle g_{-23}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1\\
-1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}}\oplus 5V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi}\oplus V_{\omega_{1}+\psi}\oplus V_{\psi}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}-\psi}\oplus V_{-\psi} \oplus V_{\omega_{1}-2\psi}\) Made total 42922 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2\) ↪ \(F^{1}_4\) 34 out of 59
Subalgebra type: \(\displaystyle A^{2}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) . Centralizer: \(\displaystyle A^{1}_2\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_2\) Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0), (0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{4}_1\) , \(\displaystyle A^{2}_2+A^{1}_2\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 6, 4): 4, (0, 0, 0, -2): 4 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{4}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{-4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2\\
-2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 8V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}} \oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}\) Made total 2597 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_2\) ↪ \(F^{1}_4\) 35 out of 59
Subalgebra type: \(\displaystyle A^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_2\): (3, 6, 8, 4): 6, (-1, -3, -2, 0): 6 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-19}+g_{-20}\), \(\displaystyle g_{6}+g_{5}\) Positive simple generators: \(\displaystyle g_{20}+g_{19}\), \(\displaystyle g_{-5}+g_{-6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\) Made total 25529 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_2\) ↪ \(F^{1}_4\) 36 out of 59
Subalgebra type: \(\displaystyle A^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_2\): (3, 6, 8, 4): 6, (0, -3, -4, -2): 6 Dimension of subalgebra generated by predefined or computed generators: 8. Negative simple generators: \(\displaystyle g_{-14}+g_{-20}+g_{-22}\), \(\displaystyle -g_{16}+g_{10}+g_{9}\) Positive simple generators: \(\displaystyle g_{22}+g_{20}+g_{14}\), \(\displaystyle g_{-2}+g_{-9}+g_{-10}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 6\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus 4V_{\omega_{1}+\omega_{2}}\) Made total 4426514 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{1}_1\) ↪ \(F^{1}_4\) 37 out of 59
Subalgebra type: \(\displaystyle 3A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle 2A^{1}_1\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle 4A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 3, 4, 2): 2, \(\displaystyle A^{1}_1\): (0, 1, 2, 2): 2, \(\displaystyle A^{1}_1\): (0, 1, 2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-16}\), \(\displaystyle g_{-9}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{16}\), \(\displaystyle g_{9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{2}+2\psi}\oplus V_{\omega_{1}+2\psi} \oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi}\oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{\omega_{2}-2\psi} \oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\) Made total 450 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+2A^{1}_1\) ↪ \(F^{1}_4\) 38 out of 59
Subalgebra type: \(\displaystyle A^{2}_1+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1+A^{1}_1\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 6, 4): 4, \(\displaystyle A^{1}_1\): (1, 2, 2, 0): 2, \(\displaystyle A^{1}_1\): (1, 0, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{-14}\), \(\displaystyle g_{-1}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{14}\), \(\displaystyle g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+\omega_{3}+2\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}+\psi} \oplus V_{\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{3}-\psi} \oplus V_{\omega_{1}+\omega_{2}-\psi}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{3}-2\psi}\oplus V_{2\omega_{1}-2\psi}\) Made total 34696 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{2}_1\) ↪ \(F^{1}_4\) 39 out of 59
Subalgebra type: \(\displaystyle 3A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{2}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 6, 4): 4, \(\displaystyle A^{2}_1\): (2, 2, 2, 0): 4, \(\displaystyle A^{2}_1\): (0, 2, 2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-20}+g_{-22}\), \(\displaystyle g_{-5}+g_{-11}\), \(\displaystyle g_{-6}\) Positive simple generators: \(\displaystyle g_{22}+g_{20}\), \(\displaystyle g_{11}+g_{5}\), \(\displaystyle g_{6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 25934 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1+A^{1}_1\) ↪ \(F^{1}_4\) 40 out of 59
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 12, 8): 16, \(\displaystyle A^{3}_1\): (2, 3, 4, 0): 6, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-12}+g_{-13}\), \(\displaystyle g_{-8}-g_{-9}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle 2g_{13}+2g_{12}\), \(\displaystyle -g_{9}+g_{8}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\
0 & 2/3 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\
0 & 6 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 642 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+2A^{1}_1\) ↪ \(F^{1}_4\) 41 out of 59
Subalgebra type: \(\displaystyle A^{10}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1+A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 10, 14, 8): 20, \(\displaystyle A^{1}_1\): (0, 1, 2, 0): 2, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 9. Negative simple generators: \(\displaystyle g_{-8}+g_{-16}\), \(\displaystyle g_{-9}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle 4g_{16}+3g_{8}\), \(\displaystyle g_{9}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{2}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 541 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{2}_1\) ↪ \(F^{1}_4\) 42 out of 59
Subalgebra type: \(\displaystyle A^{1}_2+A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, \(\displaystyle A^{2}_1\): (0, 0, 2, 2): 4 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-7}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{7}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+4\psi}\oplus V_{\omega_{1}+2\omega_{3}+2\psi}\oplus V_{\omega_{3}+3\psi}\oplus V_{\omega_{2}+\omega_{3}+\psi} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{3}-\psi}\oplus V_{\omega_{2}+2\omega_{3}-2\psi} \oplus V_{0}\oplus V_{\omega_{3}-3\psi}\oplus V_{\omega_{1}-4\psi}\) Made total 446 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2+A^{8}_1\) ↪ \(F^{1}_4\) 43 out of 59
Subalgebra type: \(\displaystyle A^{1}_2+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, \(\displaystyle A^{8}_1\): (0, 0, 4, 4): 16 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-4}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle 2g_{4}+2g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+4\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\) Made total 2776 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{1}_1\) ↪ \(F^{1}_4\) 44 out of 59
Subalgebra type: \(\displaystyle B^{1}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 3, 4, 2): 2, (-2, -2, -2, 0): 4, \(\displaystyle A^{1}_1\): (0, 1, 2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 13. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{8}\), \(\displaystyle g_{-9}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-8}\), \(\displaystyle g_{9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{2}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{1}+\omega_{3}+2\psi}\oplus V_{\omega_{2}+2\psi}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\) Made total 7313 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{2}_1\) ↪ \(F^{1}_4\) 45 out of 59
Subalgebra type: \(\displaystyle B^{1}_2+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 3, 4, 2): 2, (-2, -2, -2, 0): 4, \(\displaystyle A^{2}_1\): (0, 2, 2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 13. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{11}+g_{5}\), \(\displaystyle -g_{-2}+g_{-9}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-5}+g_{-11}\), \(\displaystyle g_{9}-g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+2\omega_{3}}\oplus 2V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\) Made total 57977 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{1}_2+A^{8}_1\) ↪ \(F^{1}_4\) 46 out of 59
Subalgebra type: \(\displaystyle G^{1}_2+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle G^{1}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (3, 6, 8, 4): 6, (-1, -3, -4, -2): 2, \(\displaystyle A^{8}_1\): (0, 0, 4, 4): 16 Dimension of subalgebra generated by predefined or computed generators: 17. Negative simple generators: \(\displaystyle g_{-19}+g_{-20}\), \(\displaystyle g_{23}\), \(\displaystyle g_{-3}-g_{-4}\) Positive simple generators: \(\displaystyle g_{20}+g_{19}\), \(\displaystyle g_{-23}\), \(\displaystyle -2g_{4}+2g_{3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/4\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3 & 0\\
-3 & 2 & 0\\
0 & 0 & 16\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+4\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}}\) Made total 818444 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_1\) ↪ \(F^{1}_4\) 47 out of 59
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\) Basis of Cartan of centralizer: 1 vectors:
(1, -1, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 6, 4): 4, (0, 0, 0, -2): 4, \(\displaystyle A^{1}_1\): (1, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{4}\), \(\displaystyle g_{-5}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{5}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{3}+6\psi}\oplus V_{2\omega_{2}+4\psi}\oplus V_{2\omega_{1}+\omega_{3}+2\psi}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{2\omega_{2}+\omega_{3}-2\psi}\oplus V_{0}\oplus V_{2\omega_{1}-4\psi}\oplus V_{\omega_{3}-6\psi}\) Made total 446 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{4}_1\) ↪ \(F^{1}_4\) 48 out of 59
Subalgebra type: \(\displaystyle A^{2}_2+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 6, 4): 4, (0, 0, 0, -2): 4, \(\displaystyle A^{4}_1\): (2, 2, 0, 0): 8 Dimension of subalgebra generated by predefined or computed generators: 11. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{4}\), \(\displaystyle g_{-1}+g_{-2}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{-4}\), \(\displaystyle 2g_{2}+2g_{1}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 8\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{3}}\oplus V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 2776 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3\) ↪ \(F^{1}_4\) 49 out of 59
Subalgebra type: \(\displaystyle A^{1}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle A^{2}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1) Contained up to conjugation as a direct summand of: \(\displaystyle A^{1}_3+A^{2}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 15. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{2}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{3}}\oplus 2V_{\omega_{3}}\oplus 3V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{2}+2\psi}\oplus V_{2\psi}\oplus V_{\omega_{3}+\psi}\oplus V_{\omega_{1}+\psi}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}} \oplus V_{0}\oplus V_{\omega_{3}-\psi}\oplus V_{\omega_{1}-\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{-2\psi}\) Made total 444 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_3\) ↪ \(F^{1}_4\) 50 out of 59
Subalgebra type: \(\displaystyle B^{1}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) . Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension). The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\) Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, 0)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_3\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -2, -2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 21. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{6}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-6}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -2\\
0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi}\oplus V_{\omega_{3}+\psi}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}-\psi}\oplus V_{\omega_{1}-2\psi}\) Made total 632067 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3\) ↪ \(F^{1}_4\) 51 out of 59
Subalgebra type: \(\displaystyle C^{1}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) . Centralizer: \(\displaystyle A^{1}_1\) . The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\) Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0) Contained up to conjugation as a direct summand of: \(\displaystyle C^{1}_3+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 6, 4): 4, (0, 0, 0, -2): 4, (0, -1, -2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 21. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{4}\), \(\displaystyle g_{9}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & -1\\
0 & -1 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & -2\\
0 & -2 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 3V_{0}\) Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{-4\psi}\) Made total 446 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(4A^{1}_1\) ↪ \(F^{1}_4\) 52 out of 59
Subalgebra type: \(\displaystyle 4A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle 3A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 3, 4, 2): 2, \(\displaystyle A^{1}_1\): (0, 1, 2, 2): 2, \(\displaystyle A^{1}_1\): (0, 1, 2, 0): 2, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 12. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-16}\), \(\displaystyle g_{-9}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{16}\), \(\displaystyle g_{9}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}} \oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\) Made total 533 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+2A^{1}_1\) ↪ \(F^{1}_4\) 53 out of 59
Subalgebra type: \(\displaystyle B^{1}_2+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 3, 4, 2): 2, (-2, -2, -2, 0): 4, \(\displaystyle A^{1}_1\): (0, 1, 2, 0): 2, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 16. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{8}\), \(\displaystyle g_{-9}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-8}\), \(\displaystyle g_{9}\), \(\displaystyle g_{2}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}} \oplus V_{2\omega_{2}}\) Made total 533 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_2\) ↪ \(F^{1}_4\) 54 out of 59
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{1}_1\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 6, 4): 4, (0, 0, 0, -2): 4, \(\displaystyle A^{1}_2\): (1, 1, 0, 0): 2, (0, -1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 16. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{4}\), \(\displaystyle g_{-5}\), \(\displaystyle g_{2}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{5}\), \(\displaystyle g_{-2}\) 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: \(\displaystyle V_{2\omega_{2}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\) Made total 537 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3+A^{2}_1\) ↪ \(F^{1}_4\) 55 out of 59
Subalgebra type: \(\displaystyle A^{1}_3+A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -1, 0, 0): 2, \(\displaystyle A^{2}_1\): (0, 0, 0, 2): 4 Dimension of subalgebra generated by predefined or computed generators: 18. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-4}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}+2\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\) Made total 527 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+A^{1}_1\) ↪ \(F^{1}_4\) 56 out of 59
Subalgebra type: \(\displaystyle C^{1}_3+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 6, 4): 4, (0, 0, 0, -2): 4, (0, -1, -2, 0): 2, \(\displaystyle A^{1}_1\): (0, 1, 0, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 24. Negative simple generators: \(\displaystyle g_{-21}\), \(\displaystyle g_{4}\), \(\displaystyle g_{9}\), \(\displaystyle g_{-2}\) Positive simple generators: \(\displaystyle g_{21}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-9}\), \(\displaystyle g_{2}\) 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: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}}\) Made total 529 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(D^{1}_4\) ↪ \(F^{1}_4\) 57 out of 59
Subalgebra type: \(\displaystyle D^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle D^{1}_4\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -1, 0, 0): 2, (0, -1, -2, 0): 2 Dimension of subalgebra generated by predefined or computed generators: 28. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{2}\), \(\displaystyle g_{9}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-9}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & -1\\
0 & -1 & 2 & 0\\
0 & -1 & 0 & 2\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & -1\\
0 & -1 & 2 & 0\\
0 & -1 & 0 & 2\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{4}}\oplus V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\) Made total 529 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_4\) ↪ \(F^{1}_4\) 58 out of 59
Subalgebra type: \(\displaystyle B^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_4\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -1, 0, 0): 2, (0, 0, -2, 0): 4 Dimension of subalgebra generated by predefined or computed generators: 36. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{4}}\oplus V_{\omega_{2}}\) Made total 527 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(F^{1}_4\) ↪ \(F^{1}_4\) 59 out of 59
Subalgebra type: \(\displaystyle F^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra). Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_3\) . Centralizer: 0 The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle F^{1}_4\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -2, -2, 0): 4, (0, 0, 0, -2): 4 Dimension of subalgebra generated by predefined or computed generators: 52. Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{6}\), \(\displaystyle g_{4}\) Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-6}\), \(\displaystyle g_{-4}\) Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 1 & -1/2\\
0 & 0 & -1/2 & 1\\
\end{pmatrix}\) Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}\) Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}}\) Made total 22979 arithmetic operations while solving the Serre relations polynomial system.
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