6.34 (1978)

Let $\mathfrak{o}$ be an associative ring with identity. A system of its ideals $\mathfrak{A} = \{\mathfrak{A}_{ij} \mid i, j \in \mathbb{Z}\}$ is called a carpet of ideals if $\mathfrak{A}_{ik}\mathfrak{A}_{kj} \subseteq \mathfrak{A}_{ij}$ for all $i, j, k \in \mathbb{Z}$. If $\mathfrak{o}$ is commutative, then the set $\Gamma_n(\mathcal{A}) = \{x \in \text{SL}_n(\mathfrak{o}) \mid x_{ij} \equiv \delta_{ij} \pmod{\mathfrak{A}_{ij}}\}$ is a group, the (special) congruenz-subgroup modulo the carpet $\mathfrak{A}$ (the “carpet subgroup”). Under quite general conditions, it was proved in (Yu. I. Merzlyakov, Algebra i Logika, 3, no. 4 (1964), 49–59 (Russian); see also M. I. Kargapolov, Yu. I. Merzlyakov, Fundamentals of the Theory of Groups, 3rd Ed., Moscow, Nauka, 1982, p. 145 (Russian)) that in the groups $\text{GL}_n$ and $\text{SL}_n$ the mutual commutator subgroup of the congruenz-subgroups modulo a carpet of ideals shifted by $k$ and $l$ steps is again the congruenz-subgroup modulo the same carpet shifted by $k + l$ steps. Prove analogous theorems a) for orthogonal groups; b) for unitary groups.