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In a recent 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 ofthe form $^{\epsilon}\sigma^{\pm 1}$ where $\epsilon\in E_n(R)$. In thisarticle we prove similar results for the odd-dimensional orthogonal groups$O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,\Delta)$ underthe assumption that $R$ is commutative and $n\geq 3$. This yields new, shortproofs of the Sandwich Classification Theorems for the groups $O_{2n+1}(R)$ and$U_{2n+1}(R,\Delta)$.

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