Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]i, a,b,c,d∈G and i⩾0, then the entire derived subgroup G′ satisfies a positive law. In fact, G′ is an extension of a nilpotent group by a locally finite group of finite exponent.