Type: \(\displaystyle 3A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 3A^{1}_1\))
Simple basis: 3 vectors: (2, 2, 2, 1), (0, 2, 2, 1), (0, 0, 2, 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}: A^{1}_1
simple basis centralizer: 1 vectors: (0, 0, 0, 1)
Number of k-submodules of g: 15
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{3}}+V_{\omega_{2}+\omega_{3}}+V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+2V_{\omega_{3}}+2V_{\omega_{2}}+2V_{\omega_{1}}+3V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1)(0, 0, 0, -1)g_{-4}-2\varepsilon_{4}
Module 22(0, 0, -1, -1)(0, 0, 1, 0)g_{3}
g_{-7}		\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 31(0, 0, 0, 1)(0, 0, 0, 1)g_{4}2\varepsilon_{4}
Module 42(0, -1, -1, -1)(0, 1, 1, 0)g_{6}
g_{-9}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 52(0, 0, -1, 0)(0, 0, 1, 1)g_{7}
g_{-3}		\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
Module 62(-1, -1, -1, -1)(1, 1, 1, 0)g_{8}
g_{-11}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 72(0, -1, -1, 0)(0, 1, 1, 1)g_{9}
g_{-6}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
Module 83(0, 0, -2, -1)(0, 0, 2, 1)g_{10}
h_{4}+2h_{3}
g_{-10}		2\varepsilon_{3}
0
-2\varepsilon_{3}
Module 92(-1, -1, -1, 0)(1, 1, 1, 1)g_{11}
g_{-8}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 104(0, -1, -2, -1)(0, 1, 2, 1)g_{12}
g_{-2}
g_{2}
g_{-12}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 114(-1, -1, -2, -1)(1, 1, 2, 1)g_{13}
g_{-5}
g_{5}
g_{-13}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 123(0, -2, -2, -1)(0, 2, 2, 1)g_{14}
h_{4}+2h_{3}+2h_{2}
g_{-14}		2\varepsilon_{2}
0
-2\varepsilon_{2}
Module 134(-1, -2, -2, -1)(1, 2, 2, 1)g_{15}
g_{-1}
g_{1}
g_{-15}		\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 143(-2, -2, -2, -1)(2, 2, 2, 1)g_{16}
h_{4}+2h_{3}+2h_{2}+2h_{1}
g_{-16}		2\varepsilon_{1}
0
-2\varepsilon_{1}
Module 151(0, 0, 0, 0)(0, 0, 0, 0)h_{4}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 12
Heirs rejected due to not being maximally dominant: 1
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 1
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 2A^{1}_1
Potential Dynkin type extensions: 4A^{1}_1,