Type: \(\displaystyle B^{1}_3\) (Dynkin type computed to be: \(\displaystyle B^{1}_3\))
Simple basis: 3 vectors: (1, 2, 2, 2, 2, 2, 2, 2), (0, -1, 0, 0, 0, 0, 0, 0), (0, 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, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 2), (0, 0, 0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 56
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{2}}+10V_{\omega_{1}}+45V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1, -2, -2, -2, -2)(0, 0, 0, -1, -2, -2, -2, -2)g_{-52}-\varepsilon_{4}-\varepsilon_{5}
Module 21(0, 0, 0, -1, -1, -2, -2, -2)(0, 0, 0, -1, -1, -2, -2, -2)g_{-48}-\varepsilon_{4}-\varepsilon_{6}
Module 31(0, 0, 0, 0, -1, -2, -2, -2)(0, 0, 0, 0, -1, -2, -2, -2)g_{-44}-\varepsilon_{5}-\varepsilon_{6}
Module 41(0, 0, 0, -1, -1, -1, -2, -2)(0, 0, 0, -1, -1, -1, -2, -2)g_{-43}-\varepsilon_{4}-\varepsilon_{7}
Module 51(0, 0, 0, 0, -1, -1, -2, -2)(0, 0, 0, 0, -1, -1, -2, -2)g_{-39}-\varepsilon_{5}-\varepsilon_{7}
Module 61(0, 0, 0, -1, -1, -1, -1, -2)(0, 0, 0, -1, -1, -1, -1, -2)g_{-38}-\varepsilon_{4}-\varepsilon_{8}
Module 71(0, 0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, 0, -1, -2, -2)g_{-34}-\varepsilon_{6}-\varepsilon_{7}
Module 81(0, 0, 0, 0, -1, -1, -1, -2)(0, 0, 0, 0, -1, -1, -1, -2)g_{-33}-\varepsilon_{5}-\varepsilon_{8}
Module 91(0, 0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, 0, -1, -1, -2)g_{-28}-\varepsilon_{6}-\varepsilon_{8}
Module 101(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, -1, 0)g_{-26}-\varepsilon_{4}+\varepsilon_{8}
Module 111(0, 0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, 0, -1, -2)g_{-22}-\varepsilon_{7}-\varepsilon_{8}
Module 121(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, -1, -1, -1, 0)g_{-20}-\varepsilon_{5}+\varepsilon_{8}
Module 131(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, -1, -1, -1, 0, 0)g_{-19}-\varepsilon_{4}+\varepsilon_{7}
Module 141(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 151(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, -1, -1, 0, 0)g_{-13}-\varepsilon_{5}+\varepsilon_{7}
Module 161(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, -1, -1, 0, 0, 0)g_{-12}-\varepsilon_{4}+\varepsilon_{6}
Module 171(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 181(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 191(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 201(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 2121(-1, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}
g_{9}
g_{-62}
g_{45}
g_{-41}
g_{63}
g_{-37}
g_{-2}
g_{64}
-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}
-h_{2}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-64}
g_{2}
g_{37}
g_{-63}
g_{41}
g_{-45}
g_{62}
g_{-9}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{1}
-\varepsilon_{2}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{2}
0
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
\varepsilon_{2}
-\varepsilon_{1}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
Module 221(0, 0, 0, 1, 0, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 231(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 241(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 251(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 261(0, 0, 0, 1, 1, 0, 0, 0)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}\varepsilon_{4}-\varepsilon_{6}
Module 271(0, 0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}\varepsilon_{5}-\varepsilon_{7}
Module 281(0, 0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}\varepsilon_{6}-\varepsilon_{8}
Module 297(-1, -1, -1, -2, -2, -2, -2, -2)(1, 1, 1, 0, 0, 0, 0, 0)g_{16}
g_{-60}
g_{-58}
g_{-32}
g_{3}
g_{10}
g_{-61}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{4}
\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 301(0, 0, 0, 1, 1, 1, 0, 0)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}\varepsilon_{4}-\varepsilon_{7}
Module 311(0, 0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}\varepsilon_{5}-\varepsilon_{8}
Module 321(0, 0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 0, 1, 2)g_{22}\varepsilon_{7}+\varepsilon_{8}
Module 337(-1, -1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 1, 0, 0, 0, 0)g_{23}
g_{-57}
g_{-55}
g_{-27}
g_{11}
g_{17}
g_{-59}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 341(0, 0, 0, 1, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}\varepsilon_{4}-\varepsilon_{8}
Module 351(0, 0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 0, 1, 1, 2)g_{28}\varepsilon_{6}+\varepsilon_{8}
Module 367(-1, -1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 1, 0, 0, 0)g_{29}
g_{-54}
g_{-51}
g_{-21}
g_{18}
g_{24}
g_{-56}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 371(0, 0, 0, 0, 1, 1, 1, 2)(0, 0, 0, 0, 1, 1, 1, 2)g_{33}\varepsilon_{5}+\varepsilon_{8}
Module 381(0, 0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 0, 1, 2, 2)g_{34}\varepsilon_{6}+\varepsilon_{7}
Module 397(-1, -1, -1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 1, 0, 0)g_{35}
g_{-50}
g_{-47}
g_{-15}
g_{25}
g_{30}
g_{-53}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{7}
\varepsilon_{3}-\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 401(0, 0, 0, 1, 1, 1, 1, 2)(0, 0, 0, 1, 1, 1, 1, 2)g_{38}\varepsilon_{4}+\varepsilon_{8}
Module 411(0, 0, 0, 0, 1, 1, 2, 2)(0, 0, 0, 0, 1, 1, 2, 2)g_{39}\varepsilon_{5}+\varepsilon_{7}
Module 427(-1, -1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-46}
g_{-42}
g_{-8}
g_{31}
g_{36}
g_{-49}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{8}
\varepsilon_{3}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 431(0, 0, 0, 1, 1, 1, 2, 2)(0, 0, 0, 1, 1, 1, 2, 2)g_{43}\varepsilon_{4}+\varepsilon_{7}
Module 441(0, 0, 0, 0, 1, 2, 2, 2)(0, 0, 0, 0, 1, 2, 2, 2)g_{44}\varepsilon_{5}+\varepsilon_{6}
Module 451(0, 0, 0, 1, 1, 2, 2, 2)(0, 0, 0, 1, 1, 2, 2, 2)g_{48}\varepsilon_{4}+\varepsilon_{6}
Module 467(-1, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 2)g_{49}
g_{-36}
g_{-31}
g_{8}
g_{42}
g_{46}
g_{-40}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
\varepsilon_{8}
\varepsilon_{3}+\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 471(0, 0, 0, 1, 2, 2, 2, 2)(0, 0, 0, 1, 2, 2, 2, 2)g_{52}\varepsilon_{4}+\varepsilon_{5}
Module 487(-1, -1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 1, 2, 2)g_{53}
g_{-30}
g_{-25}
g_{15}
g_{47}
g_{50}
g_{-35}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
\varepsilon_{7}
\varepsilon_{3}+\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 497(-1, -1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 1, 2, 2, 2)g_{56}
g_{-24}
g_{-18}
g_{21}
g_{51}
g_{54}
g_{-29}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{6}
\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 507(-1, -1, -1, -1, 0, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2, 2)g_{59}
g_{-17}
g_{-11}
g_{27}
g_{55}
g_{57}
g_{-23}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 517(-1, -1, -1, 0, 0, 0, 0, 0)(1, 1, 1, 2, 2, 2, 2, 2)g_{61}
g_{-10}
g_{-3}
g_{32}
g_{58}
g_{60}
g_{-16}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{4}
\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 521(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 531(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 541(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 551(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 561(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: 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}_2
Potential Dynkin type extensions: B^{1}_3+A^{2}_1, B^{1}_3+A^{1}_1,