Let $F_n$ be the free group on $n$ generators - if it makes no difference we can just stick with $F_2$. I would like to show that the minimum length (in terms of the standard generating set) of an element in $[F_n,F_n]$ is 4. More generally, is there any lower bound for $\ell([g,h])$ given the lengths $\ell(g)$ and $\ell(h)$?
One example I am keeping in mind is $[x,yx^n] = xyx^{-1}y^{-1}$ so any sort of lower bound will probably have to maybe take some kind of minimum of $\ell(g)$ and $\ell(h)$.