A structure theorem for the elementary unimodular vector group

Given a pair of vectors $v,w\in R^{r+1}$ with $\langle v,w\rangle=v\cdot w^T=1$, A. Suslin constructed a matrix $S_r(v,w)\in Sl_{2^r}(R)$. We study the subgroup $SUm_r(R)$ generated by these matrices, and its (elementary) subgroup $EUm_r(R)$ generated by the matrices $S_r(e_1\varepsilon,e_1\varepsilon^{T^{-1}})$, for $\varepsilon\in E_{r+1}(R)$. The basic calculus for $EUm_r(R)$ is developed via a key lemma, and a fundamental property of Suslin matrices is derived.