Finite generation of Lie algebras associated with associative algebras

Let F be a field of characteristic not 2. An associative F-algebra R gives rise to the commutator Lie algebra R(-) = (R, [a, b] = ab - ba). If the algebra R is equipped with an involution * : R -> R then the space of the skew-symmetric elements K = {a is an element of R vertical bar a* = -a} is a Lie subalgebra of R(-). In this paper we find sufficient conditions for the Lie algebras [R, R] and [K, K] to be finitely generated. (C) 2014 Published by Elsevier Inc.