Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
Simple basis: 1 vectors: (1, 2, 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}: B^{1}_2
simple basis centralizer: 2 vectors: (0, 0, 1, 0), (0, 0, 0, 1)

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Number of k-submodules of g: 22
Module decomposition, fundamental coords over k: \(\displaystyle 3V_{2\omega_{1}}+8V_{\omega_{1}}+11V_{0}\)
g/k k-submodules

id	size	b\cap k-lowest weight	b\cap k-highest weight	Module basis	Weights epsilon coords
Module 1	1	(0, 0, -2, -1)	(0, 0, -2, -1)	g_{-10}	-2\varepsilon_{3}
Module 2	1	(0, 0, -1, -1)	(0, 0, -1, -1)	g_{-7}	-\varepsilon_{3}-\varepsilon_{4}
Module 3	1	(0, 0, 0, -1)	(0, 0, 0, -1)	g_{-4}	-2\varepsilon_{4}
Module 4	1	(0, 0, -1, 0)	(0, 0, -1, 0)	g_{-3}	-\varepsilon_{3}+\varepsilon_{4}
Module 5	2	(-1, -1, -2, -1)	(0, 1, 0, 0)	g_{2} g_{-13}	\varepsilon_{2}-\varepsilon_{3} -\varepsilon_{1}-\varepsilon_{3}
Module 6	1	(0, 0, 1, 0)	(0, 0, 1, 0)	g_{3}	\varepsilon_{3}-\varepsilon_{4}
Module 7	1	(0, 0, 0, 1)	(0, 0, 0, 1)	g_{4}	2\varepsilon_{4}
Module 8	2	(0, -1, -2, -1)	(1, 1, 0, 0)	g_{5} g_{-12}	\varepsilon_{1}-\varepsilon_{3} -\varepsilon_{2}-\varepsilon_{3}
Module 9	2	(-1, -1, -1, -1)	(0, 1, 1, 0)	g_{6} g_{-11}	\varepsilon_{2}-\varepsilon_{4} -\varepsilon_{1}-\varepsilon_{4}
Module 10	1	(0, 0, 1, 1)	(0, 0, 1, 1)	g_{7}	\varepsilon_{3}+\varepsilon_{4}
Module 11	2	(0, -1, -1, -1)	(1, 1, 1, 0)	g_{8} g_{-9}	\varepsilon_{1}-\varepsilon_{4} -\varepsilon_{2}-\varepsilon_{4}
Module 12	2	(-1, -1, -1, 0)	(0, 1, 1, 1)	g_{9} g_{-8}	\varepsilon_{2}+\varepsilon_{4} -\varepsilon_{1}+\varepsilon_{4}
Module 13	1	(0, 0, 2, 1)	(0, 0, 2, 1)	g_{10}	2\varepsilon_{3}
Module 14	2	(0, -1, -1, 0)	(1, 1, 1, 1)	g_{11} g_{-6}	\varepsilon_{1}+\varepsilon_{4} -\varepsilon_{2}+\varepsilon_{4}
Module 15	2	(-1, -1, 0, 0)	(0, 1, 2, 1)	g_{12} g_{-5}	\varepsilon_{2}+\varepsilon_{3} -\varepsilon_{1}+\varepsilon_{3}
Module 16	2	(0, -1, 0, 0)	(1, 1, 2, 1)	g_{13} g_{-2}	\varepsilon_{1}+\varepsilon_{3} -\varepsilon_{2}+\varepsilon_{3}
Module 17	3	(-2, -2, -2, -1)	(0, 2, 2, 1)	g_{14} g_{-1} g_{-16}	2\varepsilon_{2} -\varepsilon_{1}+\varepsilon_{2} -2\varepsilon_{1}
Module 18	3	(-1, -2, -2, -1)	(1, 2, 2, 1)	g_{15} h_{4}+2h_{3}+2h_{2}+h_{1} g_{-15}	\varepsilon_{1}+\varepsilon_{2} 0 -\varepsilon_{1}-\varepsilon_{2}
Module 19	3	(0, -2, -2, -1)	(2, 2, 2, 1)	g_{16} g_{1} g_{-14}	2\varepsilon_{1} \varepsilon_{1}-\varepsilon_{2} -2\varepsilon_{2}
Module 20	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{1}	0
Module 21	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{3}	0
Module 22	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{4}	0

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Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 3
Heirs rejected due to not being maximally dominant: 13
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 13
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, A^{2}_1+A^{1}_1, 2A^{2}_1,