This is a cross-post of my question here on MathOverflow.
Let $g$ be a nontrivial element of a finitely generated free group $G$. Is there a finite index subgroup $H \subset G$ in which $g$ is one element of a basis?
Apparently this is a special case of Marshall Hall's theorem, and there is a well-known proof of this more general result due to Stallings. However, I'm wondering if there is a more "elementary"/"self-contained" way of seeing that such a finite index subgroup exists.