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In group theory it is well-known that every subgroup of finite index contains a normal subgroup of finite index. It is not true in general that for Lie algebras every subalgebra of finite codimenslon contains an ideal of finite codimension [p. 94, I. Stewart, Lie Algebras,]. A theorem of G.P. Kukin [Problem of equality and free products of Lie algebras and associa- tive algebras, Sibirsk. Mar. Zh. 24 (1983), 85–96 ] states that any restricted subalgebra H of L of finite codimension contains an ideal of finite codimension in L.

My question is as follows:

Are there any general results in this direction? Does the notion "core of a subalgebra in a Lie algebra" plays similar roles as in groups in some cases?

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You might find the following paper interesting: MR1813893 (2001m:17024) Reviewed Riley, David(3-WON); Tasić, Vladimir(3-NB) On the growth of subalgebras in Lie p-algebras. (English summary) J. Algebra 237 (2001), no. 1, 273–286. The authors use the core notion to count subalgebras of finite codimension.

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