Sandwich classification for $O_{2n+1}(R)$ and $U_{2n+1}(R,\Delta )$ revisited

Abstract

In a recent paper, the author proved that if $n\ge 3$ is a natural number, $R$ a commutative ring and $\sigma \in G{L}_n(R)$, then $t_{kl}({\sigma }_{ij})$ where $i\ne j$ and $k\ne l$ can be expressed as a product of $8$ matrices of the form ${}^{\epsilon }{\sigma }^{\pm 1}$ where $\epsilon \in E_n(R)$. In this article we prove similar results for the odd-dimensional orthogonal groups $O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,\Delta )$ under the assumption that $R$ is commutative and $n\ge 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_{2n+1}(R)$ and $U_{2n+1}(R,\Delta )$.