I am trying to understand which $2$ by $2$ real matrices represent complex numbers in following way.

Let $J=\begin{bmatrix} 0&1\\-1&0\end{bmatrix}$ and $A=\begin{bmatrix} a&b\\c&d\end{bmatrix}$ be any real matrix.

If $A$ represents a complex matrix (by standard embedding of complex field into matrix ring) then $A$ should commute with the matrix $J$, which image of complex number $i$.

Q. I want to understand why the matrices commuting with $J$ are precisely the matrices representing complex numbers?