Commutators of congruence subgroups in the arithmetic case

In our joint paper with Alexei Stepanov it was established that for any two comaximal ideals $A$ and $B$ of a commutative ring $R$, $A+B=R$, and any $n\ge 3$ one has $[E(n,R,A),E(n,R,B)]=E(n,R,AB)$. Alec Mason and Wilson Stothers constructed counterexamples that show that the above equality may fail when $A$ and $B$ are not comaximal, even for such nice rings as $\mathbb{Z}[i]$. In the present note, we establish a rather striking result that this equality, and thus also the stronger equality $[\mathrm{GL}(n,R,A),\mathrm{GL}(n,R,B)]=E(n,R,AB)$, do hold when $R$ is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is a blend of elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and an explicit computation of multirelative $\mathrm{SK}_1$ from my 1982 paper, which in turn relied on very deep arithmetical results by Jean-Pierre Serre, and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl).