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

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Number of k-submodules of g: 16
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{2}}+4V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+2V_{\omega_{2}}+2V_{\omega_{1}}+6V_{0}\)
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, 0, -1, 0)	g_{-3}	-\varepsilon_{3}+\varepsilon_{4}
Module 2	1	(-1, 0, 0, 0)	(-1, 0, 0, 0)	g_{-1}	-\varepsilon_{1}+\varepsilon_{2}
Module 3	1	(1, 0, 0, 0)	(1, 0, 0, 0)	g_{1}	\varepsilon_{1}-\varepsilon_{2}
Module 4	1	(0, 0, 1, 0)	(0, 0, 1, 0)	g_{3}	\varepsilon_{3}-\varepsilon_{4}
Module 5	2	(0, 0, -1, -1)	(0, 0, 0, 1)	g_{4} g_{-7}	\varepsilon_{4} -\varepsilon_{3}
Module 6	2	(0, 0, 0, -1)	(0, 0, 1, 1)	g_{7} g_{-4}	\varepsilon_{3} -\varepsilon_{4}
Module 7	2	(-1, -1, -1, -1)	(0, 1, 1, 1)	g_{9} g_{-11}	\varepsilon_{2} -\varepsilon_{1}
Module 8	3	(0, 0, -1, -2)	(0, 0, 1, 2)	g_{10} 2h_{4}+h_{3} g_{-10}	\varepsilon_{3}+\varepsilon_{4} 0 -\varepsilon_{3}-\varepsilon_{4}
Module 9	2	(0, -1, -1, -1)	(1, 1, 1, 1)	g_{11} g_{-9}	\varepsilon_{1} -\varepsilon_{2}
Module 10	4	(-1, -1, -2, -2)	(0, 1, 1, 2)	g_{12} g_{-8} g_{2} g_{-15}	\varepsilon_{2}+\varepsilon_{4} -\varepsilon_{1}+\varepsilon_{4} \varepsilon_{2}-\varepsilon_{3} -\varepsilon_{1}-\varepsilon_{3}
Module 11	4	(0, -1, -2, -2)	(1, 1, 1, 2)	g_{13} g_{-6} g_{5} g_{-14}	\varepsilon_{1}+\varepsilon_{4} -\varepsilon_{2}+\varepsilon_{4} \varepsilon_{1}-\varepsilon_{3} -\varepsilon_{2}-\varepsilon_{3}
Module 12	4	(-1, -1, -1, -2)	(0, 1, 2, 2)	g_{14} g_{-5} g_{6} g_{-13}	\varepsilon_{2}+\varepsilon_{3} -\varepsilon_{1}+\varepsilon_{3} \varepsilon_{2}-\varepsilon_{4} -\varepsilon_{1}-\varepsilon_{4}
Module 13	4	(0, -1, -1, -2)	(1, 1, 2, 2)	g_{15} g_{-2} g_{8} g_{-12}	\varepsilon_{1}+\varepsilon_{3} -\varepsilon_{2}+\varepsilon_{3} \varepsilon_{1}-\varepsilon_{4} -\varepsilon_{2}-\varepsilon_{4}
Module 14	3	(-1, -2, -2, -2)	(1, 2, 2, 2)	g_{16} 2h_{4}+2h_{3}+2h_{2}+h_{1} g_{-16}	\varepsilon_{1}+\varepsilon_{2} 0 -\varepsilon_{1}-\varepsilon_{2}
Module 15	1	(0, 0, 0, 0)	(0, 0, 0, 0)	h_{1}	0
Module 16	1	(0, 0, 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: 10
Heirs rejected due to not being maximally dominant: 2
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 2
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 2
Parabolically induced by A^{1}_1
Potential Dynkin type extensions: 3A^{1}_1,