Unipotence of a double commutator with transvections

Let $R$ be a nonzero commutative ring with identity, $n\geq3$, and $σ\in GL_n(R)$. We prove that there exist nonidentity transvections $τ_1,τ_2\in GL_n(R)$ such that $\big[[σ,τ_1],τ_2\big]=I_n$. Hence, Kourovka Notebook Problem 10.46 has a stronger affirmative answer over commutative rings with iden