A note on strongly Lie nilpotency (Q810616)

Let R be an associative ring. Define the Lie multiplication on R and the Lie central series by \(R^ 1=R\) and \(R^ n=[R^{n-1},R]\). If \(R^ n=0\) for some n call R Lie nilpotent. For subsets A, B of R define \((A,B)=[A,B]R\) and consider the series \(R^{(1)}=R\) and \(R^{(n)}=(R^{(n-1)},R)\). If \(R^{(n)}=0\) for some n call R strongly Lie nilpotent. The author proves that if R is strongly Lie nilpotent then \(R^{(2)}\) is nilpotent.