So I am trying to show that the commutator subgroup of the free group with two generators $F_2=\langle a, b \rangle$ contain more than just commutators. That is, there exists $\sigma \in [F_2, F_2]$ such that $\sigma\neq \alpha \beta \alpha^{-1} \beta^{-1} \forall \alpha ,\beta \in F_2$.
One problem is given $\sigma$ it is hard to rule out all possibilities. For example, it is hard to check if $aba^{-1}b^{-1}aba^{-1}b^{-1}$ is a commutator, since \begin{align*} \alpha \beta & = \alpha\mathbf{(aa)(a^{-1}a^{-1})}\beta\\ &= \alpha(a \mathbf{b b^{-1}} a)(a^{-1}a^{-1})\beta,\\ \end{align*} which does not have the expression $\alpha c c^{-1}\beta$, which seems to imply that we cannot reduce it easily.
Any source/ reference will be helpful.