Constructing diagonalizable matricies such that matrix exponential of their product is easy to compute.

Question

Question Consider a real skew-symmetric matrix $J$ and a real symmetric positive semi-definite matrix $A$. We want to parameterize a construction of these matrices so that it is easy to compute $e^{JA}$. How can we do this?

Attempt

1. A real skew-symmetric matrix admits an Eigendecomposition as $J = U^{-1}_J D_J U_J$, where $U_J$ is a unitary matrix and $D_J$ is a diagonal matrix with purely complex values.
2. A real symmetric positive semi-definite matrix admits an Eigendecomposition as $A = U^{-1}_A D_A U_A$, where $U_A$ is an orthogonal matrix and $D_A$ is a diagonal matrix with purely real non-negative values.
3. The matrix exponential of $J$ is $e^J = U^{-1}_J e^{D_J} U_J$, where $e^{D_J}$ is just the exponentiation of the diagonals.
4. The matrix exponential of $A$ is $e^A = U^{-1}_A e^{D_A} U_A$, where $e^{D_A}$ is just the exponentiation of the diagonals.
5. We have that $e^{JA} = e^{U^{-1}_J D_J U_J U^{-1}_A D_A U_A}$. How can we simplify this?
6. What is an easy way to construct $U_J$ and $U_A$?

Remark The solution to a Hamiltonian dynamical system with a quadratic Hamiltonian is given by $x(t) = e^{JA(t - t_0)}x(t_0)$. How can we construct $J$ and $A$ so that this is easy to simplify?