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

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Number of k-submodules of g: 10
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+2V_{\omega_{2}}+2V_{\omega_{1}}+3V_{0}\)
g/k k-submodules

id	size	b\cap k-lowest weight	b\cap k-highest weight	Module basis	Weights epsilon coords
Module 1	1	(0, 0, -1)	(0, 0, -1)	g_{-3}	-2\varepsilon_{3}
Module 2	2	(0, -1, -1)	(0, 1, 0)	g_{2} g_{-5}	\varepsilon_{2}-\varepsilon_{3} -\varepsilon_{2}-\varepsilon_{3}
Module 3	1	(0, 0, 1)	(0, 0, 1)	g_{3}	2\varepsilon_{3}
Module 4	2	(-1, -1, -1)	(1, 1, 0)	g_{4} g_{-6}	\varepsilon_{1}-\varepsilon_{3} -\varepsilon_{1}-\varepsilon_{3}
Module 5	2	(0, -1, 0)	(0, 1, 1)	g_{5} g_{-2}	\varepsilon_{2}+\varepsilon_{3} -\varepsilon_{2}+\varepsilon_{3}
Module 6	2	(-1, -1, 0)	(1, 1, 1)	g_{6} g_{-4}	\varepsilon_{1}+\varepsilon_{3} -\varepsilon_{1}+\varepsilon_{3}
Module 7	3	(0, -2, -1)	(0, 2, 1)	g_{7} h_{3}+2h_{2} g_{-7}	2\varepsilon_{2} 0 -2\varepsilon_{2}
Module 8	4	(-1, -2, -1)	(1, 2, 1)	g_{8} g_{-1} g_{1} g_{-8}	\varepsilon_{1}+\varepsilon_{2} -\varepsilon_{1}+\varepsilon_{2} \varepsilon_{1}-\varepsilon_{2} -\varepsilon_{1}-\varepsilon_{2}
Module 9	3	(-2, -2, -1)	(2, 2, 1)	g_{9} h_{3}+2h_{2}+2h_{1} g_{-9}	2\varepsilon_{1} 0 -2\varepsilon_{1}
Module 10	1	(0, 0, 0)	(0, 0, 0)	h_{3}	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: 7
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 A^{1}_1
Potential Dynkin type extensions: 3A^{1}_1,