Type: \(\displaystyle D^{1}_4\) (Dynkin type computed to be: \(\displaystyle D^{1}_4\))
Simple basis: 4 vectors: (1, 2, 2, 2, 2, 2), (0, -1, 0, 0, 0, 0), (0, 0, -1, 0, 0, 0), (0, 0, -1, -2, -2, -2)
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}: B^{1}_2
simple basis centralizer: 2 vectors: (0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 16
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{2}}+5V_{\omega_{1}}+10V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, -1, -2)(0, 0, 0, 0, -1, -2)g_{-16}-\varepsilon_{5}-\varepsilon_{6}
Module 21(0, 0, 0, 0, -1, -1)(0, 0, 0, 0, -1, -1)g_{-11}-\varepsilon_{5}
Module 31(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, -1)g_{-6}-\varepsilon_{6}
Module 41(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 528(-1, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0)g_{1}
g_{7}
g_{-34}
g_{12}
g_{33}
g_{-32}
g_{-8}
g_{35}
g_{-30}
g_{-3}
g_{-2}
g_{36}
-2h_{6}-2h_{5}-2h_{4}-h_{3}
-h_{2}
-h_{3}
2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-36}
g_{2}
g_{3}
g_{30}
g_{-35}
g_{8}
g_{32}
g_{-33}
g_{-12}
g_{34}
g_{-7}
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 61(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 71(0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 1)g_{6}\varepsilon_{6}
Module 81(0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 1, 1)g_{11}\varepsilon_{5}
Module 91(0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 1, 2)g_{16}\varepsilon_{5}+\varepsilon_{6}
Module 108(-1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 0, 0)g_{17}
g_{-29}
g_{-27}
g_{-24}
g_{4}
g_{9}
g_{13}
g_{-31}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 118(-1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 0)g_{21}
g_{-26}
g_{-23}
g_{-20}
g_{10}
g_{14}
g_{18}
g_{-28}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 128(-1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1)g_{25}
g_{-22}
g_{-19}
g_{-15}
g_{15}
g_{19}
g_{22}
g_{-25}		\varepsilon_{1}
-\varepsilon_{2}
-\varepsilon_{3}
-\varepsilon_{4}
\varepsilon_{4}
\varepsilon_{3}
\varepsilon_{2}
-\varepsilon_{1}
Module 138(-1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 2)g_{28}
g_{-18}
g_{-14}
g_{-10}
g_{20}
g_{23}
g_{26}
g_{-21}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 148(-1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 2, 2)g_{31}
g_{-13}
g_{-9}
g_{-4}
g_{24}
g_{27}
g_{29}
g_{-17}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 151(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 161(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{6}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 6
Heirs rejected due to not being maximally dominant: 6
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 6
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_3
Potential Dynkin type extensions: D^{1}_4+A^{1}_1, D^{1}_4+A^{2}_1,