Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
Simple basis: 1 vectors: (1, 2, 3, 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}: A^{1}_3
simple basis centralizer: 3 vectors: (0, 1, 0, 0), (1, 0, 0, 0), (0, 1, 2, 0)
Number of k-submodules of g: 30
Module decomposition, fundamental coords over k: \(\displaystyle 7V_{2\omega_{1}}+8V_{\omega_{1}}+15V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(-1, -2, -2, 0)(-1, -2, -2, 0)g_{-14}-\varepsilon_{1}-\varepsilon_{2}
Module 21(-1, -1, -2, 0)(-1, -1, -2, 0)g_{-11}-\varepsilon_{1}-\varepsilon_{3}
Module 31(0, -1, -2, 0)(0, -1, -2, 0)g_{-9}-\varepsilon_{2}-\varepsilon_{3}
Module 41(-1, -1, 0, 0)(-1, -1, 0, 0)g_{-5}-\varepsilon_{1}+\varepsilon_{3}
Module 51(0, -1, 0, 0)(0, -1, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 61(-1, 0, 0, 0)(-1, 0, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 71(1, 0, 0, 0)(1, 0, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 81(0, 1, 0, 0)(0, 1, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 92(-1, -2, -3, -1)(0, 0, 0, 1)g_{4}
g_{-19}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 101(1, 1, 0, 0)(1, 1, 0, 0)g_{5}\varepsilon_{1}-\varepsilon_{3}
Module 112(-1, -2, -2, -1)(0, 0, 1, 1)g_{7}
g_{-17}		-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 121(0, 1, 2, 0)(0, 1, 2, 0)g_{9}\varepsilon_{2}+\varepsilon_{3}
Module 132(-1, -1, -2, -1)(0, 1, 1, 1)g_{10}
g_{-15}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 141(1, 1, 2, 0)(1, 1, 2, 0)g_{11}\varepsilon_{1}+\varepsilon_{3}
Module 152(0, -1, -2, -1)(1, 1, 1, 1)g_{12}
g_{-13}		1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 162(-1, -1, -1, -1)(0, 1, 2, 1)g_{13}
g_{-12}		-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 171(1, 2, 2, 0)(1, 2, 2, 0)g_{14}\varepsilon_{1}+\varepsilon_{2}
Module 182(0, -1, -1, -1)(1, 1, 2, 1)g_{15}
g_{-10}		1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 193(-2, -3, -4, -2)(0, 1, 2, 2)g_{16}
g_{-8}
g_{-24}		-\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}
-\varepsilon_{1}+\varepsilon_{4}
Module 202(0, 0, -1, -1)(1, 2, 2, 1)g_{17}
g_{-7}		1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 213(-1, -3, -4, -2)(1, 1, 2, 2)g_{18}
g_{-6}
g_{-23}		-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{4}
Module 222(0, 0, 0, -1)(1, 2, 3, 1)g_{19}
g_{-4}		1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 233(-1, -2, -4, -2)(1, 2, 2, 2)g_{20}
g_{-3}
g_{-22}		-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{4}
Module 243(-1, -2, -3, -2)(1, 2, 3, 2)g_{21}
2h_{4}+3h_{3}+2h_{2}+h_{1}
g_{-21}		-\varepsilon_{4}
0
\varepsilon_{4}
Module 253(-1, -2, -2, -2)(1, 2, 4, 2)g_{22}
g_{3}
g_{-20}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{3}
\varepsilon_{3}+\varepsilon_{4}
Module 263(-1, -1, -2, -2)(1, 3, 4, 2)g_{23}
g_{6}
g_{-18}		\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}
\varepsilon_{2}+\varepsilon_{4}
Module 273(0, -1, -2, -2)(2, 3, 4, 2)g_{24}
g_{8}
g_{-16}		\varepsilon_{1}-\varepsilon_{4}
\varepsilon_{1}
\varepsilon_{1}+\varepsilon_{4}
Module 281(0, 0, 0, 0)(0, 0, 0, 0)h_{1}0
Module 291(0, 0, 0, 0)(0, 0, 0, 0)h_{2}0
Module 301(0, 0, 0, 0)(0, 0, 0, 0)h_{3}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 7
Heirs rejected due to not being maximally dominant: 18
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 18
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 0
Potential Dynkin type extensions: A^{2}_2, B^{2}_2, 2A^{2}_1, A^{2}_1+A^{1}_1,