Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
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}_3
simple basis centralizer: 3 vectors: (0, 0, 1, 0), (0, 1, 0, 0), (0, 0, 1, 2)

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Number of k-submodules of g: 22
Module decomposition, fundamental coords over k: \(\displaystyle 7V_{2\omega_{1}}+15V_{0}\)
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, -2, -2)	(0, -1, -2, -2)	g_{-14}	-\varepsilon_{2}-\varepsilon_{3}
Module 2	1	(0, -1, -1, -2)	(0, -1, -1, -2)	g_{-12}	-\varepsilon_{2}-\varepsilon_{4}
Module 3	1	(0, 0, -1, -2)	(0, 0, -1, -2)	g_{-10}	-\varepsilon_{3}-\varepsilon_{4}
Module 4	1	(0, -1, -1, 0)	(0, -1, -1, 0)	g_{-6}	-\varepsilon_{2}+\varepsilon_{4}
Module 5	1	(0, 0, -1, 0)	(0, 0, -1, 0)	g_{-3}	-\varepsilon_{3}+\varepsilon_{4}
Module 6	1	(0, -1, 0, 0)	(0, -1, 0, 0)	g_{-2}	-\varepsilon_{2}+\varepsilon_{3}
Module 7	3	(-1, -2, -2, -2)	(1, 0, 0, 0)	g_{1} g_{-9} g_{-16}	\varepsilon_{1}-\varepsilon_{2} -\varepsilon_{2} -\varepsilon_{1}-\varepsilon_{2}
Module 8	1	(0, 1, 0, 0)	(0, 1, 0, 0)	g_{2}	\varepsilon_{2}-\varepsilon_{3}
Module 9	1	(0, 0, 1, 0)	(0, 0, 1, 0)	g_{3}	\varepsilon_{3}-\varepsilon_{4}
Module 10	3	(-1, -1, -2, -2)	(1, 1, 0, 0)	g_{5} g_{-7} g_{-15}	\varepsilon_{1}-\varepsilon_{3} -\varepsilon_{3} -\varepsilon_{1}-\varepsilon_{3}
Module 11	1	(0, 1, 1, 0)	(0, 1, 1, 0)	g_{6}	\varepsilon_{2}-\varepsilon_{4}
Module 12	3	(-1, -1, -1, -2)	(1, 1, 1, 0)	g_{8} g_{-4} g_{-13}	\varepsilon_{1}-\varepsilon_{4} -\varepsilon_{4} -\varepsilon_{1}-\varepsilon_{4}
Module 13	1	(0, 0, 1, 2)	(0, 0, 1, 2)	g_{10}	\varepsilon_{3}+\varepsilon_{4}
Module 14	3	(-1, -1, -1, -1)	(1, 1, 1, 1)	g_{11} h_{4}+h_{3}+h_{2}+h_{1} g_{-11}	\varepsilon_{1} 0 -\varepsilon_{1}
Module 15	1	(0, 1, 1, 2)	(0, 1, 1, 2)	g_{12}	\varepsilon_{2}+\varepsilon_{4}
Module 16	3	(-1, -1, -1, 0)	(1, 1, 1, 2)	g_{13} g_{4} g_{-8}	\varepsilon_{1}+\varepsilon_{4} \varepsilon_{4} -\varepsilon_{1}+\varepsilon_{4}
Module 17	1	(0, 1, 2, 2)	(0, 1, 2, 2)	g_{14}	\varepsilon_{2}+\varepsilon_{3}
Module 18	3	(-1, -1, 0, 0)	(1, 1, 2, 2)	g_{15} g_{7} g_{-5}	\varepsilon_{1}+\varepsilon_{3} \varepsilon_{3} -\varepsilon_{1}+\varepsilon_{3}
Module 19	3	(-1, 0, 0, 0)	(1, 2, 2, 2)	g_{16} g_{9} g_{-1}	\varepsilon_{1}+\varepsilon_{2} \varepsilon_{2} -\varepsilon_{1}+\varepsilon_{2}
Module 20	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{2}	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: 7
Heirs rejected due to not being maximally dominant: 11
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 11
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, 2A^{2}_1, A^{2}_1+A^{1}_1,