TITLE: Proof that every three-dimensional Einstein manifold has constant curvature
QUESTION [5 upvotes]: In pseudo-Riemannian geometry it is well known that every three-dimensional Einstein manifold has constant curvature. A proof of this is sketched here.
Question. Does anyone know where in the literature I can find a proof of such result?

REPLY [6 votes]: This is precisely Proposition 1.120 on p.49 in Besse's "Einstein Manifolds" (I am using the reprint of the 1987 edition, so the numbering may be different in the older edition):

A 3-dimensional pseudo-Riemannian manifold is Einstein iff it has  constant (sectional) curvature.

Note that the proof is before the proposition.