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

Abstract: In a paper, the author proved that if $n\geq 3$ is a natural number, $R$ a commutative ring and $\sigma\in GL_n(R)$, then $t_{kl}(\sigma_{ij})$ where $i\neq j$ and $k\neq l$ 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 $n\geq 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_{2n+1}(R)$ and $U_{2n+1}(R,\Delta)$.