Semisimple weakly symmetric pseudo-Riemannian manifolds

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature $$(n-1,1)$$(n-1,1) and trans-Lorentzian signature $$(n-2,2)$$(n-2,2).