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Let  n  be a natural number greater than or equal to 3,  R  a commutative ring andσ  ∈    GL  n   ⁢   (  R  ){\sigma\in\mathrm{GL}_{n}(R)}   . We show thatt   k  ⁢  l    ⁢   (   σ   i  ⁢  j    ){t_{kl}(\sigma_{ij})}    (resp.t   k  ⁢  l    ⁢   (    σ   i  ⁢  i    -   σ   j  ⁢  j){t_{kl}(\sigma_{ii}-\sigma_{jj})}   ), wherei  ≠  j{i\neq j}    andk  ≠  l{k\neq l}    can be expressed as a product of eight (resp. 24) matrices of the formσ±  1ϵ{{}^{\epsilon}\sigma^{\pm 1}}   , whereϵ  ∈    E  n   ⁢   (  R  ){\epsilon\in E_{n}(R)}   . We prove similar results for the orthogonal groupsO   2  ⁢  n    ⁢   (  R  ){\mathrm{O}_{2n}(R)}    and the hyperbolic unitary groupsU   2  ⁢  n    ⁢   (  R  ,  Λ  ){\mathrm{U}_{2n}(R,\Lambda)}    under the assumption that  R  is commutative andn  ≥  3{n\geq 3}   . This yields new, very short proofs of the Sandwich Classification Theorems for the groupsGL  n   ⁢   (  R  ){\mathrm{GL}_{n}(R)}   ,O   2  ⁢  n    ⁢   (  R  ){\mathrm{O}_{2n}(R)}    andU   2  ⁢  n    ⁢   (  R  ,  Λ  ){\mathrm{U}_{2n}(R,\Lambda)}   .

Sandwich classification for GL n ( R ), O 2 n ( R ) and U 2 n ( R ,Λ) revisited