While working on a problem in quantum optics, I came across the following determinant of a complex matrix of size $n \times n$ :
$$\mathbb{G}=\det\left[\mathcal{U}_{11}^{}+\mathcal{U}_{12}^{}\mathcal{D}_{S}^{}\right]$$
where $\mathcal{U}_{xy}^{}$ are blocks of a $2 \times 2$ block partitioned complex Symplectic matrix defined as,
$$\begin{pmatrix}\mathcal{U}_{11}^{} & \mathcal{U}_{12}^{} \\ \mathcal{U}_{21}^{} & \mathcal{U}_{22}^{}\end{pmatrix}:=\mathcal{U}_{2n \times 2n}^{}=e_{}^{-\mathcal{H}^{}\Sigma_{}^{}}$$
with $\mathcal{H}^{}\Sigma_{}^{}$ being a complex Hamiltonian matrix expressed as a product of complex $2 \times 2$ block partitioned symmetric matrix $$\mathcal{H}^{}=\begin{pmatrix}\mathcal{A}_{n \times n}^{}=\mathcal{A}_{n \times n}^{T} & \mathcal{B}_{n \times n}^{} \\ \mathcal{B}_{n \times n}^{T} & \mathcal{D}_{n \times n}^{}=\mathcal{D}_{n \times n}^{T}\end{pmatrix}$$ and the standard symplectic matrix $$\Sigma_{}^{}=\begin{pmatrix}\mathbb{O}_{n \times n}^{} & \mathbb{I}_{n \times n}^{} \\ -\mathbb{I}_{n \times n}^{} & \mathbb{O}_{n \times n}^{}\end{pmatrix}.$$ Further $\mathcal{D}_{S}^{}$ is a $n \times n$ complex symmetric matrix and . $\mathbb{O}_{n \times n}^{}$, $\mathbb{I}_{n \times n}^{}$ are respectively null and identity matrices of dimension $n \times n$.
$\mathbf{Question :}$ Is it possible to express $\mathbb{G}$ in terms of expression involving traces or determinants of sums of products of $\mathcal{A}_{n \times n}^{}$, $\mathcal{B}_{n \times n}^{}$, $\mathcal{D}_{n \times n}^{}$, $\mathcal{D}_{S}^{}$ matrices and eigenvalues of the symplectic matrix $\mathcal{H}^{}\Sigma_{}^{}$ without explicit computation of $\mathcal{U}_{2n \times 2n}^{}$, in an elegant general form? (I could do this for $n=2$ case using combination of Cayley–Hamilton theorem and Faddeev-Leverrier algorithm. For general case of $n$, I am not able to achieve this task. Without any rigorous reasoning, I suspect some elegant expression for $\mathbb{G}$ can be given. Is this feasible?)