  A method, due to \'Elie Cartan, is used to give an algebraic classification
of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four.
Only one case with Lorentz signature can be Einstein without having constant
curvature, and two cases with (2,2) signature are Einstein of which one is
Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian
manifold is simply connected, then it is shown to be diffeomorphic to
$\real^4$. All metrics for the simply connected non-reductive Einstein spaces
are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous
spaces of dimension two and none of dimension three with connected isotropy
subgroup.
