Let \(R\) be an integral domain of characteristic \(0\) and \(G\) be a group. For any normal subgroup \(M\) of \(G\) denote by \(\Delta_R(G,M)\) the kernel of the natural homomorphism from \(RG\) to \(RG/M\). For any group \(M\) let \(TM\) be the set of torsion elements in \(M\). The author proves that if \(N\) is a nilpotent and \(A\) is an Abelian subgroup of \(G\) so that \(A\leq N\) and that \(N/TN\) is finitely generated, then the group \(\{u\in U(RG)\mid u-1\in\Delta_R(G,N)\cdot\Delta_R(G,A)\}\) is torsion-free. This generalizes an earlier theorem of the author. The assumption on \(N/TN\) is necessary. The author gives an example where the statement fails without that hypothesis.