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?