 Solution

Suppose $C=S{\Lambda }_C{S}^{-1}$ and $D=S{\Lambda }_D{S}^{-1}$ for an invertible $S$ and diagonal matrices ${\Lambda }_C,{\Lambda }_D$. Since diagonal matrices commute,Therefore simultaneous diagonalization implies commutation. This argument works over either $\mathbb{R}$ or $\mathbb{C}$ and explains why simultaneous diagonalization produces commuting operators.