Examples about that $\exp(X+Y)=\exp(X) \exp(Y)$ does not imply $[X,Y]=0$ where $X,Y$ are $n \times n $ matrix

Question

I read the https://en.wikipedia.org/wiki/Matrix_exponential

There is a saying that "The converse is not true in general. The equation $\exp(X+Y)=\exp(X) \exp(Y)$ does not imply that X and Y commute."

I would like to know some concrete examples.

Answer 1

Hint

Consider $X=\begin{pmatrix} \pi i&0\\0&-\pi i\end{pmatrix}$ and $Y=\begin{pmatrix} 0&1\\0&-2\pi i\end{pmatrix}.$