Type: \(\displaystyle A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_1\))
Simple basis: 1 vectors: (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^{3}_1
simple basis centralizer: 1 vectors: (1, 0)

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Number of k-submodules of g: 8
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+4V_{\omega_{1}}+3V_{0}\)
g/k k-submodules

id	size	b\cap k-lowest weight	b\cap k-highest weight	Module basis	Weights epsilon coords
Module 1	1	(-1, 0)	(-1, 0)	g_{-1}	-\varepsilon_{1}+\varepsilon_{2}
Module 2	1	(1, 0)	(1, 0)	g_{1}	\varepsilon_{1}-\varepsilon_{2}
Module 3	2	(-3, -1)	(0, 1)	g_{2} g_{-5}	-2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3} -\varepsilon_{1}+2\varepsilon_{2}-\varepsilon_{3}
Module 4	2	(-2, -1)	(1, 1)	g_{3} g_{-4}	-\varepsilon_{1}+\varepsilon_{3} \varepsilon_{2}-\varepsilon_{3}
Module 5	2	(-1, -1)	(2, 1)	g_{4} g_{-3}	-\varepsilon_{2}+\varepsilon_{3} \varepsilon_{1}-\varepsilon_{3}
Module 6	2	(0, -1)	(3, 1)	g_{5} g_{-2}	\varepsilon_{1}-2\varepsilon_{2}+\varepsilon_{3} 2\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}
Module 7	3	(-3, -2)	(3, 2)	g_{6} 2h_{2}+3h_{1} g_{-6}	-\varepsilon_{1}-\varepsilon_{2}+2\varepsilon_{3} 0 \varepsilon_{1}+\varepsilon_{2}-2\varepsilon_{3}
Module 8	1	(0, 0)	(0, 0)	h_{1}	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: 5
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 0
Potential Dynkin type extensions: A^{1}_2, 2A^{1}_1, G^{1/3}_2,