= Simple spectrum gives simultaneous diagonalization
Suppose $D$ has $n$ distinct <eigenvalues> over a field and $C$ commutes with $D$. Each <eigenspace> of $D$ is one-dimensional. Since <commuting maps preserve eigenspaces>, $C$ acts as a scalar on each of them. The <eigenvectors> of $D$ give a common <eigenbasis> and hence <simultaneous diagonalization>. For real matrices, the result gives a real <eigenbasis> only if the <eigenvalues> of $D$ are real; otherwise it applies over $\mathbb C$.
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