Powers of commutators in linear algebraic groups

Let $\mathcal{G}$ be a linear algebraic group over $k$, where $k$ is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let $G=\mathcal{G}(k)$. We prove that if $\gamma ,\delta \in G$ such that $\gamma$ is a commutator and $⟨\delta ⟩=⟨\gamma ⟩$ then $\delta$ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.