Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension

Let M be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group G acts transitively and isometrically on M. In this situation, the metric on M induces a bilinear form 〈⋅,⋅〉 on the Lie algebra g of G which is nil-invariant, a property closely related to invariance. We study such spaces M in three important cases. First, we assume 〈⋅,⋅〉 is invariant, in which case the Einstein property requires that G is either solvable or semisimple. Next, we investigate the case where G is solvable. Here, M is compact and M=G/Γ for a lattice Γ in G. We show that in dimensions less or equal to 7, compact quotients M=G/Γ exist only for nilpotent groups G. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. Finally, we consider semisimple Lie groups G, and find that M splits as a pseudo-Riemannian product of Einstein quotients for the compact and the non-compact factors of G.