Let $F$ be a free group generated by two elements. Let $\{a,b\}$ and $\{c,d\}$ are two different generating set.
Q:Prove that $[a,b]$ is either conjugate to $[c,d]$ or its inverse.
Here $[a,b]=aba^{-1}b^{-1}$ is the commutator.
If we consider $F$ to be the fundamental group of torus with one puncture then I have a geometric proof. It will be better if I can have an algebraic proof for this complete algebraic fact.