  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
$\langle\cdot,\cdot\rangle$ on the Lie algebra $\mathfrak{g}$ of $G$ which is
nil-invariant, a property closely related to invariance. We study such spaces
$M$ in three important cases. First, we assume $\langle\cdot,\cdot\rangle$ 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 $M=G/\Gamma$ for a lattice $\Gamma$ in $G$. We show
that in dimensions less or equal to $7$, compact quotients $M=G/\Gamma$ 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
$M$ is covered by a pseudo-Riemannian product of Einstein manifolds
corresponding to the compact and the non-compact factors of $G$.
