Let F be a free group, and let \gamma_n(F) be the n-th term of the lower central series of F. We prove that F/[\gamma_j(F),\gamma_i(F),\gamma_k(F)] and F/[\gamma_j(F), \gamma_i(F), \gamma_k(F), \gamma_l(F)] are torsion free and residually nilpotent for certain values of i,j,k and i,j,k,l, respectively. In the process of proving this, we prove that the analogous Lie rings are torsion free.