Let $ A \in \Bbb R^{2n \times 2n}$ and let the matrix exponential $ e^A $ be defined by
$$ e^A := \sum_{k=0}^{\infty}\frac{A^k}{k!}.$$
I want to ask under what conditions the matrix $ e^A $ is symplectic, i.e.,
$$ (e^A)^TJe^A=J,$$
where $$ J=\left(\begin{matrix} 0 & I_n\\ -I_n& 0 \end{matrix}\right). $$
I have known that if $ B $ is symmetric, then $$ (e^{JB})^TJe^{JB}=J. $$ However, it seems that I cannot apply this to the original problem. Can you give me some hints or referneces?