Free groups of outer commutator varieties of groups, J. London Math. Soc. (2) 64 (2001), 423-435.

Abstract In the previous paper, the author proved that if F is a free group, 1 < i <= j <= 2i and i \leq k <= i+j+1 then F/[\gamma_j(F),\gamma_i(F),\gamma_k(F)] is residually nilpotent and torsion-free. In this article, we extend this result to 1 < i <= j \leq 2i and i \leq k \leq 2i + 2j. As in the previous paper, we prove that the analogous Lie rings, L/[L^j,L^i,L^k] where L is a free Lie ring, are torsion-free. We also find candidates for torsion in L/[L^J,L^i,L^k] whenever k is the least of { i,j,k}, and prove the existence of torsion in L/[L^j,L^i,L^k] when i,j,k <= 5 and k is the least of { i,j,k}.