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

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Number of k-submodules of g: 16
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+6V_{\omega_{1}}+9V_{0}\)
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, -1, -1, 0)	(0, -1, -1, 0)	g_{-6}	-\varepsilon_{2}+\varepsilon_{4}
Module 2	1	(0, 0, -1, 0)	(0, 0, -1, 0)	g_{-3}	-\varepsilon_{3}+\varepsilon_{4}
Module 3	1	(0, -1, 0, 0)	(0, -1, 0, 0)	g_{-2}	-\varepsilon_{2}+\varepsilon_{3}
Module 4	2	(0, -1, -1, -1)	(1, 0, 0, 0)	g_{1} g_{-9}	\varepsilon_{1}-\varepsilon_{2} -\varepsilon_{2}+\varepsilon_{5}
Module 5	1	(0, 1, 0, 0)	(0, 1, 0, 0)	g_{2}	\varepsilon_{2}-\varepsilon_{3}
Module 6	1	(0, 0, 1, 0)	(0, 0, 1, 0)	g_{3}	\varepsilon_{3}-\varepsilon_{4}
Module 7	2	(-1, -1, -1, 0)	(0, 0, 0, 1)	g_{4} g_{-8}	\varepsilon_{4}-\varepsilon_{5} -\varepsilon_{1}+\varepsilon_{4}
Module 8	2	(0, 0, -1, -1)	(1, 1, 0, 0)	g_{5} g_{-7}	\varepsilon_{1}-\varepsilon_{3} -\varepsilon_{3}+\varepsilon_{5}
Module 9	1	(0, 1, 1, 0)	(0, 1, 1, 0)	g_{6}	\varepsilon_{2}-\varepsilon_{4}
Module 10	2	(-1, -1, 0, 0)	(0, 0, 1, 1)	g_{7} g_{-5}	\varepsilon_{3}-\varepsilon_{5} -\varepsilon_{1}+\varepsilon_{3}
Module 11	2	(0, 0, 0, -1)	(1, 1, 1, 0)	g_{8} g_{-4}	\varepsilon_{1}-\varepsilon_{4} -\varepsilon_{4}+\varepsilon_{5}
Module 12	2	(-1, 0, 0, 0)	(0, 1, 1, 1)	g_{9} g_{-1}	\varepsilon_{2}-\varepsilon_{5} -\varepsilon_{1}+\varepsilon_{2}
Module 13	3	(-1, -1, -1, -1)	(1, 1, 1, 1)	g_{10} h_{4}+h_{3}+h_{2}+h_{1} g_{-10}	\varepsilon_{1}-\varepsilon_{5} 0 -\varepsilon_{1}+\varepsilon_{5}
Module 14	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{2}	0
Module 15	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{3}	0
Module 16	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{4}-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: 1
Heirs rejected due to not being maximally dominant: 10
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 10
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,