Spectrum of tridiagonal block matrix

Question

I have the following $4n \times 4n$ block tridiagonal matrix:

$$ \begin{bmatrix} M_{1} && -M_{2} && 0 && \cdots&&&& &&0 \\ M_{2} && M_{1} && -JI_{2} &&\cdots&&&&&& 0\\ 0&& JI_{2} && 0 && -hI_{2} &&&& && \vdots\\ 0&& 0 && hI_{2} && 0 &&\ddots&& &&\\ \vdots && &&&& \ddots&&\ddots && -JI_{2} && 0\\ 0 && && &&&&JI_{2}&& M_{1}^{'}&&-M_{2}^{'}\\ 0 && 0 &&&& \cdots && 0&&M_{2}^{'} &&M_{1}^{'} \end{bmatrix} $$

Each block is a $2\times2$ matrix with complex entries and $I_{2}$ is the $2\times2$ identity matrix and $0\neq h,J \in \mathbb{R}$ and I know the following:

1. $M_{1},M_{1}^{'}$ are diagonal matrices
2. $M_{2}$ and $M_{2}^{'}$ are of the form $\begin{bmatrix} h && * \\ 0 && h\end{bmatrix}$

My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?