Suppose $G = F/R$ is a finitely presented group with $F = F_n$ a free group and $R$ the normal closure of words $W_1, \dots, W_p$, not all of which are (products of) commutators. An obvious necessary condition for $G$ to admit a presentation with commutator relators is that the abelianization be torsion-free; then the number of generators in a commutator-relators presentation will be equal to the rank of $G^{ab}$. This certainly isn't sufficient, as a nontrivial perfect group shows. Is there a criterion for deciding whether a given presentation can be changed by Tietze moves into one with commutator relators? Is there an algorithm for doing so?

EDIT: Andy Putman observes that a general algorithm would solve the problem of identifying the trivial group, so there can't be such a criterion. Given this, are there any known results for special classes of presentations?