Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that:

$e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$

Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x i\sigma_x\otimes\sigma_x + J_y i\sigma_x\otimes\sigma_y + J_z i\sigma_z\otimes\sigma_z$ for all $J_k$ real and non-zero, $\sigma_k$ the Pauli matrices and $b=i\sigma_z\otimes I$, consider the function $F:\mathbb{R}^2\rightarrow SU(4)$ defined by:

$F(w_1,w_2)=e^{a+w_1b}e^{a+w_2b}$

Is it actually possible to find some suitable $U,V$ in this case in terms of $a,b$ and $w_1,w_2$ and thus write $F$ explicitly in the form of the RHS of Thompson's formula? If so, how can I do this without simply trying to explicitly evaluate the exponentials.