I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where,

$J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$

$A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$.

$B \in \mathbb{C}^{n \times n}$ and is ${\bf symmetric}$ so its conjugate $\bar{B}= B^*$.

These properties make $J$ Hermitian and thus with real eigenvalues.

Now I want to separate the $n$ largest eigenvalues of $J$ in a way that can be expressed as eigenvalues of an $n \times n$ matrix constructed based on $A$ and $B$. The ultimate goal is to separate the positive and negative eigenvalues of $J$ when there are exactly $n$ positive and $n$ negative eigenvalues and identify the cases when this symmetry goes away by only inspecting the $n \times n$ matrices (functions of $A$ and $B$).

I already know that if $K=\begin{pmatrix} A&B \\B & A \end{pmatrix}$ (notice that no restriction is required on $A$ or $B$) the e-values of $K$ is the union of the e-values of $A+B$ and $A-B$. Can there be any similar formulation for $J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$ or $J=\begin{pmatrix} A&B \\\bar{B} & \bar{A} \end{pmatrix}$

I would appreciate any help with the above problem.