``Hasse principle'' for \(\text{GL}_2(D)\) (Q1969188)

Let \(D\) be a Euclidean domain, and let \(G=\text{GL}_2(D)\). The author shows that every automorphism of \(G\) that preserves conjugacy classes is an inner automorphism. It follows that every cocycle \(f\colon G\to G\) (\(G\) acting on itself by conjugation) that is locally a coboundary (i.e., it becomes a coboundary when restricted to any cyclic subgroup of \(G\)) is already a coboundary. The author calls this the ``Hasse principle'' for \(G\). To see the implication (which is stated without reference in the note), note that for a local coboundary \(f\), the map \(x\mapsto f(x)x\) is an automorphism of the required type.