On finite separability of relatively free Lie algebras (Q1333644)

Let \(F\) be a field of characteristic zero. Suppose that \(L\) is a free Lie algebra in a variety \(V\) of Lie algebras defined by identities \[ \begin{aligned} &(\text{ad }x) (\text{ad }y)^ n = \sum_{j=1}^ n \alpha_ j (\text{ad }y)^ j (\text{ad } x) (\text{ad }y)^{n-j},\\ &(\text{ad} [x_ 1, y_ 1])\cdots (\text{ad} [x_{m-1}, y_{m-1}]) (\text{ad } x_ m)=0, \end{aligned} \] where \(\alpha_ j\in F\). Then \(L\) is finitely separable that is for any finitely generated subalgebra \(H\) in \(L\) and any element \(x\in L\setminus H\) there exists an ideal \(I\) in \(L\) of a finite codimension such that \(x\in L\setminus (I+H)\). These identities characterize locally residually finite varieties of Lie algebras that is varieties of Lie algebras in which each finitely generated algebra is residually finite dimensional [see the author, Mat. Zametki 44, 352--361 (1988; Zbl 0662.17012)].