Lemma 32.5:

Suppose $G$ is a group and $x_1, x_2,\dots, x_k$ are distinct elements of $G$ such that $x_i \neq x_j^{-1}$ if $i\neq j$. Let $w$ be a word involving integer powers of $x_1, x_2,\dots, x_k$ such that the sum of the exponents of each $x_i$ is zero. Then $w$ lies in the commutator subgroup of $G$.