I would like to use the following lemma.
Let $F$ be a free group and let $X=\lbrace x,y \rbrace$, where $x,y \in F$. Put $G=\langle x,y \rangle$. If $G$ is not free over $X$, then $x$ and $y$ commute.
I suppose it holds and I guess I could prove it using the Nielsen-Schreier theorem. However, the statement seems really trivial, so I suppose it must have been already formulated or there must be some more general theorem from which it immediately follows. On the other hand, I tried to google something like this and failed. So, could you please point me to some book/paper where it can be found or provide me with some nice proof (hopefully not much technical)?