Sandwich classification for $G{L}_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,\Lambda )$ revisited

Abstract

Let $n$ be a natural number greater or equal to $3$, $R$ a commutative ring and $\sigma \in G{L}_n(R)$. We show that $t_{kl}({\sigma }_{ij})$ (resp. $t_{kl}({\sigma }_{ii}-{\sigma }_{jj}))$ where $i\ne j$ and $k\ne l$ 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 $n\ge 3$. This yields new, very short proofs of the Sandwich Classification Theorems for the groups $G{L}_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,\Lambda )$.