Type: \(\displaystyle 0\) (Dynkin type computed to be: \(\displaystyle 0\))
Simple basis: 0 vectors:
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}: E^{1}_6
simple basis centralizer: 6 vectors: (0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 78
Module decomposition, fundamental coords over k: \(\displaystyle 78V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(-1, -2, -2, -3, -2, -1)(-1, -2, -2, -3, -2, -1)g_{-36}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 21(-1, -1, -2, -3, -2, -1)(-1, -1, -2, -3, -2, -1)g_{-35}-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 31(-1, -1, -2, -2, -2, -1)(-1, -1, -2, -2, -2, -1)g_{-34}-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 41(-1, -1, -1, -2, -2, -1)(-1, -1, -1, -2, -2, -1)g_{-33}1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 51(-1, -1, -2, -2, -1, -1)(-1, -1, -2, -2, -1, -1)g_{-32}-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 61(0, -1, -1, -2, -2, -1)(0, -1, -1, -2, -2, -1)g_{-31}\varepsilon_{4}+\varepsilon_{5}
Module 71(-1, -1, -1, -2, -1, -1)(-1, -1, -1, -2, -1, -1)g_{-30}1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 81(-1, -1, -2, -2, -1, 0)(-1, -1, -2, -2, -1, 0)g_{-29}-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 91(0, -1, -1, -2, -1, -1)(0, -1, -1, -2, -1, -1)g_{-28}\varepsilon_{3}+\varepsilon_{5}
Module 101(-1, -1, -1, -1, -1, -1)(-1, -1, -1, -1, -1, -1)g_{-27}1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 111(-1, -1, -1, -2, -1, 0)(-1, -1, -1, -2, -1, 0)g_{-26}1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 121(0, -1, -1, -1, -1, -1)(0, -1, -1, -1, -1, -1)g_{-25}\varepsilon_{2}+\varepsilon_{5}
Module 131(-1, 0, -1, -1, -1, -1)(-1, 0, -1, -1, -1, -1)g_{-24}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 141(0, -1, -1, -2, -1, 0)(0, -1, -1, -2, -1, 0)g_{-23}\varepsilon_{3}+\varepsilon_{4}
Module 151(-1, -1, -1, -1, -1, 0)(-1, -1, -1, -1, -1, 0)g_{-22}1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 161(0, 0, -1, -1, -1, -1)(0, 0, -1, -1, -1, -1)g_{-21}-\varepsilon_{1}+\varepsilon_{5}
Module 171(0, -1, 0, -1, -1, -1)(0, -1, 0, -1, -1, -1)g_{-20}\varepsilon_{1}+\varepsilon_{5}
Module 181(0, -1, -1, -1, -1, 0)(0, -1, -1, -1, -1, 0)g_{-19}\varepsilon_{2}+\varepsilon_{4}
Module 191(-1, 0, -1, -1, -1, 0)(-1, 0, -1, -1, -1, 0)g_{-18}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 201(-1, -1, -1, -1, 0, 0)(-1, -1, -1, -1, 0, 0)g_{-17}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 211(0, 0, 0, -1, -1, -1)(0, 0, 0, -1, -1, -1)g_{-16}-\varepsilon_{2}+\varepsilon_{5}
Module 221(0, 0, -1, -1, -1, 0)(0, 0, -1, -1, -1, 0)g_{-15}-\varepsilon_{1}+\varepsilon_{4}
Module 231(0, -1, 0, -1, -1, 0)(0, -1, 0, -1, -1, 0)g_{-14}\varepsilon_{1}+\varepsilon_{4}
Module 241(0, -1, -1, -1, 0, 0)(0, -1, -1, -1, 0, 0)g_{-13}\varepsilon_{2}+\varepsilon_{3}
Module 251(-1, 0, -1, -1, 0, 0)(-1, 0, -1, -1, 0, 0)g_{-12}-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 261(0, 0, 0, 0, -1, -1)(0, 0, 0, 0, -1, -1)g_{-11}-\varepsilon_{3}+\varepsilon_{5}
Module 271(0, 0, 0, -1, -1, 0)(0, 0, 0, -1, -1, 0)g_{-10}-\varepsilon_{2}+\varepsilon_{4}
Module 281(0, 0, -1, -1, 0, 0)(0, 0, -1, -1, 0, 0)g_{-9}-\varepsilon_{1}+\varepsilon_{3}
Module 291(0, -1, 0, -1, 0, 0)(0, -1, 0, -1, 0, 0)g_{-8}\varepsilon_{1}+\varepsilon_{3}
Module 301(-1, 0, -1, 0, 0, 0)(-1, 0, -1, 0, 0, 0)g_{-7}-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 311(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, -1)g_{-6}-\varepsilon_{4}+\varepsilon_{5}
Module 321(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{3}+\varepsilon_{4}
Module 331(0, 0, 0, -1, 0, 0)(0, 0, 0, -1, 0, 0)g_{-4}-\varepsilon_{2}+\varepsilon_{3}
Module 341(0, 0, -1, 0, 0, 0)(0, 0, -1, 0, 0, 0)g_{-3}-\varepsilon_{1}+\varepsilon_{2}
Module 351(0, -1, 0, 0, 0, 0)(0, -1, 0, 0, 0, 0)g_{-2}\varepsilon_{1}+\varepsilon_{2}
Module 361(-1, 0, 0, 0, 0, 0)(-1, 0, 0, 0, 0, 0)g_{-1}1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}-1/2\varepsilon_{7}+1/2\varepsilon_{8}
Module 371(1, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0)g_{1}-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 381(0, 1, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0)g_{2}-\varepsilon_{1}-\varepsilon_{2}
Module 391(0, 0, 1, 0, 0, 0)(0, 0, 1, 0, 0, 0)g_{3}\varepsilon_{1}-\varepsilon_{2}
Module 401(0, 0, 0, 1, 0, 0)(0, 0, 0, 1, 0, 0)g_{4}\varepsilon_{2}-\varepsilon_{3}
Module 411(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{3}-\varepsilon_{4}
Module 421(0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 1)g_{6}\varepsilon_{4}-\varepsilon_{5}
Module 431(1, 0, 1, 0, 0, 0)(1, 0, 1, 0, 0, 0)g_{7}1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 441(0, 1, 0, 1, 0, 0)(0, 1, 0, 1, 0, 0)g_{8}-\varepsilon_{1}-\varepsilon_{3}
Module 451(0, 0, 1, 1, 0, 0)(0, 0, 1, 1, 0, 0)g_{9}\varepsilon_{1}-\varepsilon_{3}
Module 461(0, 0, 0, 1, 1, 0)(0, 0, 0, 1, 1, 0)g_{10}\varepsilon_{2}-\varepsilon_{4}
Module 471(0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 1, 1)g_{11}\varepsilon_{3}-\varepsilon_{5}
Module 481(1, 0, 1, 1, 0, 0)(1, 0, 1, 1, 0, 0)g_{12}1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 491(0, 1, 1, 1, 0, 0)(0, 1, 1, 1, 0, 0)g_{13}-\varepsilon_{2}-\varepsilon_{3}
Module 501(0, 1, 0, 1, 1, 0)(0, 1, 0, 1, 1, 0)g_{14}-\varepsilon_{1}-\varepsilon_{4}
Module 511(0, 0, 1, 1, 1, 0)(0, 0, 1, 1, 1, 0)g_{15}\varepsilon_{1}-\varepsilon_{4}
Module 521(0, 0, 0, 1, 1, 1)(0, 0, 0, 1, 1, 1)g_{16}\varepsilon_{2}-\varepsilon_{5}
Module 531(1, 1, 1, 1, 0, 0)(1, 1, 1, 1, 0, 0)g_{17}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 541(1, 0, 1, 1, 1, 0)(1, 0, 1, 1, 1, 0)g_{18}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 551(0, 1, 1, 1, 1, 0)(0, 1, 1, 1, 1, 0)g_{19}-\varepsilon_{2}-\varepsilon_{4}
Module 561(0, 1, 0, 1, 1, 1)(0, 1, 0, 1, 1, 1)g_{20}-\varepsilon_{1}-\varepsilon_{5}
Module 571(0, 0, 1, 1, 1, 1)(0, 0, 1, 1, 1, 1)g_{21}\varepsilon_{1}-\varepsilon_{5}
Module 581(1, 1, 1, 1, 1, 0)(1, 1, 1, 1, 1, 0)g_{22}-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 591(0, 1, 1, 2, 1, 0)(0, 1, 1, 2, 1, 0)g_{23}-\varepsilon_{3}-\varepsilon_{4}
Module 601(1, 0, 1, 1, 1, 1)(1, 0, 1, 1, 1, 1)g_{24}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 611(0, 1, 1, 1, 1, 1)(0, 1, 1, 1, 1, 1)g_{25}-\varepsilon_{2}-\varepsilon_{5}
Module 621(1, 1, 1, 2, 1, 0)(1, 1, 1, 2, 1, 0)g_{26}-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 631(1, 1, 1, 1, 1, 1)(1, 1, 1, 1, 1, 1)g_{27}-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 641(0, 1, 1, 2, 1, 1)(0, 1, 1, 2, 1, 1)g_{28}-\varepsilon_{3}-\varepsilon_{5}
Module 651(1, 1, 2, 2, 1, 0)(1, 1, 2, 2, 1, 0)g_{29}1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 661(1, 1, 1, 2, 1, 1)(1, 1, 1, 2, 1, 1)g_{30}-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 671(0, 1, 1, 2, 2, 1)(0, 1, 1, 2, 2, 1)g_{31}-\varepsilon_{4}-\varepsilon_{5}
Module 681(1, 1, 2, 2, 1, 1)(1, 1, 2, 2, 1, 1)g_{32}1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 691(1, 1, 1, 2, 2, 1)(1, 1, 1, 2, 2, 1)g_{33}-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 701(1, 1, 2, 2, 2, 1)(1, 1, 2, 2, 2, 1)g_{34}1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 711(1, 1, 2, 3, 2, 1)(1, 1, 2, 3, 2, 1)g_{35}1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 721(1, 2, 2, 3, 2, 1)(1, 2, 2, 3, 2, 1)g_{36}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}
Module 731(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{1}0
Module 741(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{2}0
Module 751(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{3}0
Module 761(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{4}0
Module 771(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 781(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: 0
Heirs rejected due to not being maximally dominant: 71
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 71
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
This subalgebra is not parabolically induced by anyone
Potential Dynkin type extensions: A^{1}_1,