Hello? I have a simple question about combinatorial group theory.

For a group $G$, I saw $[G_k, G_m] \subset G_{k+m}$ and these two subgroups need not be equal. Then is there any known condition that they can be equal? How about the case that $G$ is (f.g.) free?

Actually, my first question was that: for a f.g. free group $F$, $F_m \subset [F_k,F_k]$ for some big $m$?