Sandwich classification for O₂ₙ₊₁(R) and U₂ₙ₊₁(R,Δ) revisited

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In a recent paper, the author proved that if n≥ 3 is a natural number, R a commutative ring and σ∈ GLₙ(R), then tkl(σij) where i≠ j and k≠ l can be expressed as a product of 8 matrices of the form εσ± ¹ where ε∈ Eₙ(R). In this article we prove similar results for the odd-dimensional orthogonal groups O₂ₙ₊₁(R) and the odd-dimensional unitary groups U₂ₙ₊₁(R,Δ) under the assumption that R is commutative and n≥ 3. This yields new, short proofs of the Sandwich Classification Theorems for the groups O₂ₙ₊₁(R) and U₂ₙ₊₁(R,Δ).