A Lie algebra \(L\) over a field \(\mathbb F\) is an \(\mathbb F\)-vector space together with a bilinear map \([\;,\;] : L \times L \to L\), denoted by \((x, y) \mapsto [x, y]\) and called the bracket of \(x\) and \(y\) such that the following axioms are satisfied:

(i)

\([x, x] = 0\),

(ii)

\( [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0\) (Jacobi identity),

for every \(x, y, z\) in \(L\).
Any associative \(\mathbb F\)-algebra \(R\) gives rise to:

\(\bullet\)

A Lie algebra \(R^{-}\) with Lie bracket \([x,y]:=xy-yx\), for all \(x, y \in R\).

\(\bullet\)

If the algebra \(R\) is equipped with an involution \(* :R\to R\) then the space of the skew-symmetric elements \(K=Skew(R,*)\) can be seen as a subalgebra of the Lie algebra \(R^-\).

In the paper under review the authors proved that if \(R\) is a finitely generated associative \(\mathbb F\)-algebra with an idempotent \(e\) such that \(ReR=R(1-e)R=R\), then the Lie algebra \([R, R]\) is finitely generated. Moreover, if \(R\) is a \(\mathbb F\)-algebra with an involution \(* :R\to R\) and an idempotent \(e\) such that \(ee^*=e^*e=0\) and \(ReR=R(1-e-e^*)R=R\), then the Lie algebra \([K, K]\) is finitely generated. In both cases the authors also show that the idempotent condition can not be dropped.