Simplification of determinant of block matrix

Question

For the following 4x4 matrix $M$, \begin{equation} \mathbf{M} = \begin{pmatrix} a & -b & 0 & d\\ -b & a & -d & 0\\ 0 & -d & -a & -b \\ d & 0 & -b & -a \\ \end{pmatrix} \end{equation} it can be expressed in 2x2 block matrix form so that, \begin{equation} \mathbf{M} = \begin{pmatrix} A & B\\ -B & D \end{pmatrix} \end{equation} Using the well known formula, \begin{equation} \begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(A-BD^{-1}C)\det(D) \end{equation} The determinant of of $M$ can be \begin{equation} \begin{pmatrix} A & B\\ -B & D \end{pmatrix}=\det(A+BD^{-1}B)\det(D) \end{equation} Can it be simplified further?