The \(E_ 2(R)\)-normalized subgroups of \(\text{GL}_ 2(R)\). II (Q1904229)

Let \(R\) be an arithmetic Dedekind domain with infinitely many units and let \(E_2(R)\) be the subgroup of \(\text{GL}_2(R)\) generated by the elementary matrices. A subgroup \(S\) of \(\text{GL}_2(R)\) is called standard if \(\text{GL}_2'({\mathbf q})\geq S\geq E_2({\mathbf q})\), for some \(R\)-ideal \({\mathbf q}\), where (i) \(E_2({\mathbf q})\) is the normal subgroup of \(E_2(R)\) generated by the \({\mathbf q}\)-elementary matrices, and (ii) \(\text{GL}_2'({\mathbf q})\) is the set of those matrices in \(\text{GL}_2(R)\) which are scalar, \((\text{mod }{\mathbf q})\). In part I [ibid. 172, No. 2, 584-611 (1995; Zbl 0827.20058)]\ the author proved that two arithmetic conditions on \(R\) are necessary and sufficient to ensure that the standard subgroups of \(\text{GL}_2(R)\) are precisely those normalized by \(E_2(R)\) with the exception of the case where \(R\) is the ring of integers of a totally imaginary number field whose only roots of unity are \(\pm 1\). Here the author extends the result to include this case.