Finite generation of Lie algebras associated to associative algebras

Abstract: 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 $:R\rightarrow R$ then the space of the skew-symmetric elements $K={a \in R \mid a^{}=-a }$ 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.