  We devise a spectral divide-and-conquer scheme for matrices that are
self-adjoint with respect to a given indefinite scalar product (i.e.
pseudosymmetic matrices). The pseudosymmetric structure of the matrix is
preserved in the spectral division, such that the method can be applied
recursively to achieve full diagonalization. The method is well-suited for
structured matrices that come up in computational quantum physics and
chemistry. In this application context, additional definiteness properties
guarantee a convergence of the matrix sign function iteration within two steps
when Zolotarev functions are used. The steps are easily parallelizable.
Furthermore, it is shown that the matrix decouples into symmetric definite
eigenvalue problems after just one step of spectral division.
