Complex product structures on some simple Lie groups

We construct invariant complex product (hyperparacomplex, indefinite quaternion) structures on the manifolds underlying the real noncompact simple Lie groups $SL(2m-1,\text{RR})$, $SU(m,m-1)$ and $SL(2m-1,\text{CC}{)}^{\text{RR}}$. We show that on the last two series of groups some of these structures are compatible with the biinvariant Killing metric. Thus we also provide a class of examples of compact (neutral) hyperparahermitean, non-flat Einstein manifolds.