Geodesic
Relative decomposition of transvections: explicit bounds
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 38, Tome 513 (2022), pp. 9-21

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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 $E(n,R,I)$. 
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     author = {M. A. Buryakov and N. A. Vavilov},
     title = {Relative decomposition of transvections: explicit bounds},
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}
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M. A. Buryakov; N. A. Vavilov. Relative decomposition of transvections: explicit bounds. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 38, Tome 513 (2022), pp. 9-21. http://geodesic.mathdoc.fr/item/ZNSL_2022_513_a1/