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We classify four-dimensional homogeneous manifolds satisfying some generalized Einstein conditions.

It is shown that locally conformally flat weakly-Einstein manifolds are either locally symmetric, and hence a product $$N_1^m(c)\times N_2^m(-c)$$ , or otherwise they are locally homothetic to some specific warped product metrics $${\mathcal {I}}\times _fN(c)$$ . As an application we classify weakly-Einstein hypersurfaces in the Euclidean space.

Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.