Locally solvable finitary Lie algebras

A Lie subalgebra L of gl(V) is finitary if L consists of elements of finite rank. We show here that an irreducible finitary algebra with a non-zero ab elian ideal is finite dimensional. Additional results require that for some positive integer d, the finitary Lie algebra L be d-bounded; i.e., L has a generating set consisting of elements of rank at most d. In particular, we show that if L is an irreducible, finitary, d-bounded Lie subalgebra of gl (V) and the locally solvable radical Is(L) is non-zero, then L is finite di mensional. If, in addition, L is locally solvable, then L is solvable of de rived length at most 64d(2).