  We construct invariant complex product (hyperparacomplex, indefinite
quaternion) structures on the manifolds underlying the real noncompact simple
Lie groups $SL(2m-1,\RR)$, $SU(m,m-1)$ and $SL(2m-1,\CC)^\RR$. 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.
