On strongly inert subalgebras of an infinite-dimensional Lie algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim (A + [A, L])/A < ∞. We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A < ∞. We show that the condition of local finiteness of L is essential in this statement. © 2002 Plenum Publishing Corporation.