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)R(−)=(R,[a,b]=ab−ba). If the algebra R   is equipped with an involution ⁎:R→R⁎:R→R then the space of the skew-symmetric elements K={a∈R|a⁎=−a}K={a∈R|a⁎=−a} is a Lie subalgebra of R(−)R(−). In this paper we find sufficient conditions for the Lie algebras [R,R][R,R] and [K,K][K,K] to be finitely generated.