Unipotence of a double commutator with transvections

We prove the statement that for every $\sigma \in G{L}_n(\mathbb{R})$ with $n\ge 3$, there exist transvections ${\tau }_1,{\tau }_2\in G{L}_n(\mathbb{R})$ such that ${\left(\right[[\sigma ,{\tau }_1],{\tau }_2]-I)}^2=0$. In particular, $[[\sigma ,{\tau }_1],{\tau }_2]$ is unipotent. The proof relies on the existence of a one- or two-dimensional invariant subspace for a real endomorphism.