I want to find the eigenvectors and eigenvalues of the following $2L \times 2L$ (assume $L$ is even) block tridiagonal matrix, $$ \begin{pmatrix} R_{\phantom{1}} & R_{1} & 0 & 0 & 0 & 0 &\cdots &0\\ R_{1} & R_{\phantom{1}} & R_{2} & 0 & 0 & 0 & \cdots &0\\ 0 & R_{2} & R_{\phantom{1}} & R_{1} & 0 & 0 &\cdots & 0\\ 0 & 0 & R_{1} & R_{\phantom{1}} & R_{2} & 0 &\cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \cdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & R_{1} & R_{\phantom{1}} \end{pmatrix} $$ where $R = ih\sigma^{y}$, $R_{1} = \frac{i}{2}(-I_{2}+\gamma \sigma^{z})$, $R_{2} = \frac{i}{2}(-I_{2}-\gamma \sigma^{z}) $ , $h$ and $\gamma$ are real, $I_{n}$ is the $n \times n$ identity matrix, and $\sigma^{y}$ and $\sigma^{z}$ are pauli matrices.
I can see that the matrix is anti-hermitian so the eigenvalues are purely imaginary and I know that $R_{1}$ and $R_{2}$ commute. I am unsure of how to proceed from here