$\begin{aligned}
& A = \sum_k (b_k + i c_k)|a_k \rangle \langle a_k| \\
& A = \sum_k b_k |a_k \rangle \langle a_k| + i c_k |a_k \rangle \langle a_k| \\
& A = B + iC \\
\end{aligned}$

$B, C$ are simultaneously diagonalizable in the eigenbasis $\{ |a_k \rangle \}$
and hence $[B, C] = 0$.