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Use a reduced sequence in an HNN extension for $\gamma$. By repeatedly conjugating a prefix to the other end and reducing any resulting pinch, one obtains a cyclically reduced sequence in an HNN extension: either an element of the base group, or a sequence of positive stable-letter length with no pinch even across its cyclic junction. For example, after conjugating away the initial base coefficient, write it as $t^{{ϵ}_1}{g}_1\cdots t^{{ϵ}_k}{g}_k$. A pinch across the junction can occur only if ${ϵ}_k=-{ϵ}_1$ and $g_k\in C_{{ϵ}_k}$. Moving the first stable letter to the end then exposes that pinch and reduces the stable-letter length by two. Iteration must terminate.

If the resulting cyclically reduced sequence has $k>0$, every positive power remains reduced and has stable-letter length equal to that power times $k$. By Britton's lemma, none is the identity. Thus a finite-order element must be conjugate into the base group $G$. Since $G$ embeds in the HNN extension, the order of a base-group element is unchanged, and conjugation also preserves order. ThereforeThe argument also applies to several stable letters: cyclic pinches must involve a stable letter and its own inverse.