This may well have been asked already, but I couldn't find a link.
If $A$, $B$ are subgroups of $G$, given generating sets for $A$ and $B$, is there a "nice" generating set for $[A,B]$?
I'm aware relations with commutators are rarely as nice as you'd hope for - hoping there's still a nice answer.
If $H = \langle X \rangle$ and $K = \langle Y \rangle$, then $$[H,K] = \left\langle [x,y]^{hk} : x \in X, y \in Y, h \in H, k \in K \right\rangle.$$
This is an exercise in Suzuki, Group Theory II, Chapter 4.