Sandwich classification for $GL_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,Λ)$ revisited

Abstract: Let $n$ be a natural number greater or equal to $3$, $R$ a commutative ring and $\sigma\in GL_n(R)$. We show that $t_{kl}(\sigma_{ij})$ (resp. $t_{kl}(\sigma_{ii}-\sigma_{jj}))$ where $i\neq j$ and $k\neq l$ can be expressed as a product of $8$ (resp. $24$) matrices of the form $^{\epsilon}\sigma^{\pm 1}$ where $\epsilon\in E_n(R)$. We prove similar results for the orthogonal groups $O_{2n}(R)$ and the hyperbolic unitary groups $U_{2n}(R,\Lambda)$ under the assumption that $R$ is commutative and $n\geq 3$. This yields new, very short proofs of the Sandwich Classification Theorems for the groups $GL_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,\Lambda)$.