Relative decomposition of transvections: explicit bounds

Let $R$ be a commutative associative ring with $1$, and let $G=\mathrm{GL}(n,R)$ be the general linear group of degree $n\ge 3$ over $R$. Further, let $I\unlhd R$ be an ideal of $R$. In the present note, which is a marginalia to the paper of Alexei Stepanov and the second named author(2000), we obtain explicit expressions of the elementary transvection $gt_{ij}(\xi)g^{-1}$, where $1\le i\neq j\le n$, $\xi\in I$ and $g\in G$, as products of the Stein–Tits–Vaserstein generators of the relative elementary group $E(n,R,I)$.