Type: \(\displaystyle B^{1}_2\) (Dynkin type computed to be: \(\displaystyle B^{1}_2\))
Simple basis: 2 vectors: (1, 2, 2, 2, 2, 2, 2), (0, -1, -1, -1, -1, -1, -1)
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}: D^{1}_5
simple basis centralizer: 5 vectors: (0, 0, 0, 1, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 2), (0, 0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 56
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{2}}+10V_{\omega_{1}}+45V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, -1, -2, -2, -2, -2)(0, 0, -1, -2, -2, -2, -2)g_{-43}-\varepsilon_{3}-\varepsilon_{4}
Module 21(0, 0, -1, -1, -2, -2, -2)(0, 0, -1, -1, -2, -2, -2)g_{-40}-\varepsilon_{3}-\varepsilon_{5}
Module 31(0, 0, 0, -1, -2, -2, -2)(0, 0, 0, -1, -2, -2, -2)g_{-37}-\varepsilon_{4}-\varepsilon_{5}
Module 41(0, 0, -1, -1, -1, -2, -2)(0, 0, -1, -1, -1, -2, -2)g_{-36}-\varepsilon_{3}-\varepsilon_{6}
Module 51(0, 0, 0, -1, -1, -2, -2)(0, 0, 0, -1, -1, -2, -2)g_{-33}-\varepsilon_{4}-\varepsilon_{6}
Module 61(0, 0, -1, -1, -1, -1, -2)(0, 0, -1, -1, -1, -1, -2)g_{-32}-\varepsilon_{3}-\varepsilon_{7}
Module 71(0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, -1, -2, -2)g_{-29}-\varepsilon_{5}-\varepsilon_{6}
Module 81(0, 0, 0, -1, -1, -1, -2)(0, 0, 0, -1, -1, -1, -2)g_{-28}-\varepsilon_{4}-\varepsilon_{7}
Module 91(0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, -1, -1, -2)g_{-24}-\varepsilon_{5}-\varepsilon_{7}
Module 101(0, 0, -1, -1, -1, -1, 0)(0, 0, -1, -1, -1, -1, 0)g_{-22}-\varepsilon_{3}+\varepsilon_{7}
Module 111(0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, -1, -2)g_{-19}-\varepsilon_{6}-\varepsilon_{7}
Module 121(0, 0, 0, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, 0)g_{-17}-\varepsilon_{4}+\varepsilon_{7}
Module 131(0, 0, -1, -1, -1, 0, 0)(0, 0, -1, -1, -1, 0, 0)g_{-16}-\varepsilon_{3}+\varepsilon_{6}
Module 141(0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, -1, -1, 0)g_{-12}-\varepsilon_{5}+\varepsilon_{7}
Module 151(0, 0, 0, -1, -1, 0, 0)(0, 0, 0, -1, -1, 0, 0)g_{-11}-\varepsilon_{4}+\varepsilon_{6}
Module 161(0, 0, -1, -1, 0, 0, 0)(0, 0, -1, -1, 0, 0, 0)g_{-10}-\varepsilon_{3}+\varepsilon_{5}
Module 171(0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, -1, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 181(0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, -1, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 191(0, 0, 0, -1, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 201(0, 0, -1, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 2110(-1, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0)g_{1}
g_{34}
g_{-31}
g_{49}
-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}
2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-49}
g_{31}
g_{-34}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}
-\varepsilon_{2}
\varepsilon_{1}+\varepsilon_{2}
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}
-\varepsilon_{1}+\varepsilon_{2}
Module 221(0, 0, 1, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 231(0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 241(0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 1, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 251(0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 1, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 265(-1, -1, -2, -2, -2, -2, -2)(1, 1, 0, 0, 0, 0, 0)g_{8}
g_{-47}
g_{-27}
g_{2}
g_{-48}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 271(0, 0, 1, 1, 0, 0, 0)(0, 0, 1, 1, 0, 0, 0)g_{10}\varepsilon_{3}-\varepsilon_{5}
Module 281(0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 1, 1, 0, 0)g_{11}\varepsilon_{4}-\varepsilon_{6}
Module 291(0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 1, 1, 0)g_{12}\varepsilon_{5}-\varepsilon_{7}
Module 305(-1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 0, 0, 0, 0)g_{14}
g_{-45}
g_{-23}
g_{9}
g_{-46}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 311(0, 0, 1, 1, 1, 0, 0)(0, 0, 1, 1, 1, 0, 0)g_{16}\varepsilon_{3}-\varepsilon_{6}
Module 321(0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 0)g_{17}\varepsilon_{4}-\varepsilon_{7}
Module 331(0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 1, 2)g_{19}\varepsilon_{6}+\varepsilon_{7}
Module 345(-1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 0, 0, 0)g_{20}
g_{-42}
g_{-18}
g_{15}
g_{-44}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 351(0, 0, 1, 1, 1, 1, 0)(0, 0, 1, 1, 1, 1, 0)g_{22}\varepsilon_{3}-\varepsilon_{7}
Module 361(0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 1, 1, 2)g_{24}\varepsilon_{5}+\varepsilon_{7}
Module 375(-1, -1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 0, 0)g_{25}
g_{-39}
g_{-13}
g_{21}
g_{-41}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 381(0, 0, 0, 1, 1, 1, 2)(0, 0, 0, 1, 1, 1, 2)g_{28}\varepsilon_{4}+\varepsilon_{7}
Module 391(0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 1, 2, 2)g_{29}\varepsilon_{5}+\varepsilon_{6}
Module 405(-1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 0)g_{30}
g_{-35}
g_{-7}
g_{26}
g_{-38}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 411(0, 0, 1, 1, 1, 1, 2)(0, 0, 1, 1, 1, 1, 2)g_{32}\varepsilon_{3}+\varepsilon_{7}
Module 421(0, 0, 0, 1, 1, 2, 2)(0, 0, 0, 1, 1, 2, 2)g_{33}\varepsilon_{4}+\varepsilon_{6}
Module 431(0, 0, 1, 1, 1, 2, 2)(0, 0, 1, 1, 1, 2, 2)g_{36}\varepsilon_{3}+\varepsilon_{6}
Module 441(0, 0, 0, 1, 2, 2, 2)(0, 0, 0, 1, 2, 2, 2)g_{37}\varepsilon_{4}+\varepsilon_{5}
Module 455(-1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 2)g_{38}
g_{-26}
g_{7}
g_{35}
g_{-30}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 461(0, 0, 1, 1, 2, 2, 2)(0, 0, 1, 1, 2, 2, 2)g_{40}\varepsilon_{3}+\varepsilon_{5}
Module 475(-1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 2, 2)g_{41}
g_{-21}
g_{13}
g_{39}
g_{-25}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 481(0, 0, 1, 2, 2, 2, 2)(0, 0, 1, 2, 2, 2, 2)g_{43}\varepsilon_{3}+\varepsilon_{4}
Module 495(-1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2)g_{44}
g_{-15}
g_{18}
g_{42}
g_{-20}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 505(-1, -1, -1, 0, 0, 0, 0)(1, 1, 1, 2, 2, 2, 2)g_{46}
g_{-9}
g_{23}
g_{45}
g_{-14}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 515(-1, -1, 0, 0, 0, 0, 0)(1, 1, 2, 2, 2, 2, 2)g_{48}
g_{-2}
g_{27}
g_{47}
g_{-8}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{3}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 521(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 531(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 541(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 551(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 561(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{7}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 11
Heirs rejected due to not being maximally dominant: 39
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 39
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_1
Potential Dynkin type extensions: B^{1}_2+A^{1}_1, B^{1}_2+A^{2}_1,