Type: \(\displaystyle D^{1}_4+A^{2}_1+2A^{1}_1\) (Dynkin type computed to be: \(\displaystyle D^{1}_4+A^{2}_1+2A^{1}_1\))
Simple basis: 7 vectors: (1, 2, 2, 2, 2, 2, 2, 2), (0, -1, 0, 0, 0, 0, 0, 0), (0, 0, -1, 0, 0, 0, 0, 0), (0, 0, -1, -2, -2, -2, -2, -2), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 0, 1, 2, 2), (0, 0, 0, 0, 0, 1, 0, 0)
Simple basis epsilon form:
Simple basis epsilon form with respect to k:
Number of outer autos with trivial action on orthogonal complement and extending to autos of ambient algebra: 0
Number of outer autos with trivial action on orthogonal complement: 0.
C(k_{ss})_{ss}: 0
simple basis centralizer: 0 vectors:
Number of k-submodules of g: 14
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{5}+\omega_{6}+\omega_{7}}+V_{\omega_{1}+\omega_{6}+\omega_{7}}+V_{\omega_{1}+2\omega_{5}}+V_{2\omega_{7}}+2V_{\omega_{6}+\omega_{7}}+V_{2\omega_{6}}+3V_{2\omega_{5}}+V_{\omega_{2}}+2V_{\omega_{1}}+V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 128(-1, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}
g_{9}
g_{-62}
g_{16}
g_{61}
g_{-60}
g_{-10}
g_{63}
g_{-58}
g_{-3}
g_{-2}
g_{64}
-2h_{8}-2h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}
-h_{2}
-h_{3}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-64}
g_{2}
g_{3}
g_{58}
g_{-63}
g_{10}
g_{60}
g_{-61}
g_{-16}
g_{62}
g_{-9}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{2}
0
0
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
Module 23(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}
h_{6}
g_{-6}		\varepsilon_{6}-\varepsilon_{7}
0
-\varepsilon_{6}+\varepsilon_{7}
Module 34(0, 0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}
g_{-22}
g_{7}
g_{-28}		\varepsilon_{6}-\varepsilon_{8}
-\varepsilon_{7}-\varepsilon_{8}
\varepsilon_{7}-\varepsilon_{8}
-\varepsilon_{6}-\varepsilon_{8}
Module 43(0, 0, 0, 0, -1, -1, -1, -2)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}
g_{-8}
g_{-33}		\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{8}
-\varepsilon_{5}-\varepsilon_{8}
Module 53(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, 1, 1, 1, 1)g_{27}
h_{8}+h_{7}+h_{6}+h_{5}
g_{-27}		\varepsilon_{5}
0
-\varepsilon_{5}
Module 64(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, 1, 1, 2)g_{28}
g_{-7}
g_{22}
g_{-14}		\varepsilon_{6}+\varepsilon_{8}
-\varepsilon_{7}+\varepsilon_{8}
\varepsilon_{7}+\varepsilon_{8}
-\varepsilon_{6}+\varepsilon_{8}
Module 73(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, 1, 1, 1, 2)g_{33}
g_{8}
g_{-20}		\varepsilon_{5}+\varepsilon_{8}
\varepsilon_{8}
-\varepsilon_{5}+\varepsilon_{8}
Module 83(0, 0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, 0, 1, 2, 2)g_{34}
2h_{8}+2h_{7}+h_{6}
g_{-34}		\varepsilon_{6}+\varepsilon_{7}
0
-\varepsilon_{6}-\varepsilon_{7}
Module 98(-1, -1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-46}
g_{-42}
g_{-38}
g_{26}
g_{31}
g_{36}
g_{-49}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{4}-\varepsilon_{8}
\varepsilon_{4}-\varepsilon_{8}
\varepsilon_{3}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 1012(0, 0, 0, 0, -1, -2, -2, -2)(0, 0, 0, 0, 1, 2, 2, 2)g_{44}
g_{21}
g_{13}
g_{39}
g_{-5}
g_{-15}
g_{15}
g_{5}
g_{-39}
g_{-13}
g_{-21}
g_{-44}		\varepsilon_{5}+\varepsilon_{6}
\varepsilon_{6}
\varepsilon_{5}-\varepsilon_{7}
\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{6}
-\varepsilon_{7}
\varepsilon_{7}
\varepsilon_{5}-\varepsilon_{6}
-\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{6}
-\varepsilon_{5}-\varepsilon_{6}
Module 118(-1, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 2)g_{49}
g_{-36}
g_{-31}
g_{-26}
g_{38}
g_{42}
g_{46}
g_{-40}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
\varepsilon_{4}+\varepsilon_{8}
\varepsilon_{3}+\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 1232(-1, -1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 1, 2, 2, 2)g_{56}
g_{-24}
g_{35}
g_{53}
g_{-18}
g_{-50}
g_{-30}
g_{29}
g_{-12}
g_{48}
g_{-47}
g_{-25}
g_{-54}
g_{51}
g_{-43}
g_{19}
g_{-19}
g_{43}
g_{-51}
g_{54}
g_{25}
g_{47}
g_{-48}
g_{12}
g_{-29}
g_{30}
g_{50}
g_{18}
g_{-53}
g_{-35}
g_{24}
g_{-56}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{7}
\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{7}
\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{7}
\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 1324(-1, -1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 1, 2, 2, 2, 2)g_{59}
g_{-17}
g_{45}
g_{-11}
g_{-41}
g_{23}
g_{-4}
g_{52}
g_{-37}
g_{-57}
g_{55}
g_{-32}
g_{32}
g_{-55}
g_{57}
g_{37}
g_{-52}
g_{4}
g_{-23}
g_{41}
g_{11}
g_{-45}
g_{17}
g_{-59}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{1}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{3}
-\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{1}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 141(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{8}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 13
Heirs rejected due to not being maximally dominant: 0
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 0
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by D^{1}_4+A^{2}_1+A^{1}_1
Potential Dynkin type extensions: D^{1}_4+A^{2}_1+3A^{1}_1,