Let $S \leq G$ and $N \lhd G$ two subgroups of $G$.
Is the commutator $[SN,SN]$ equal to $[S,S]N$ ?
The first is generated by commutators $[ax,by]$ (where $a,b\in S$ and $x,y \in N$), and each generator (after inserting $e$'s in the forms $cc^{-1}$ or $c^{-1}c$ in the product $(ax)^{-1}(by)^{-1}axby$ and using the fact that $N$ is normal) can be expressed as $[a,b]n$ for a suitable $n$. Is this correct?