If \(D\subseteq \operatorname{GL}_n\) is diagonal, the coordinate functions on \(D\) are generated by restrictions of diagonal characters. These characters form a group, and the coordinate algebra decomposes into their weight spaces; in fact the nonzero weight spaces are one-dimensional.

A character basis of \(k[D]\) makes every comodule over \(k[D]\) split into weight spaces, because the coaction of a finite-dimensional representation involves only finitely many basis characters. This gives complete reducibility into one-dimensional representations.

Conversely, apply the representation criterion to a faithful finite-dimensional representation of \(D\). It decomposes as a direct sum of characters, so \(D\) embeds into the diagonal subgroup of \(\operatorname{GL}(V)\).