Four-dimensional pseudo-Riemannian homogeneous spaces whose isotropy is non-trivial with commuting curvature operators have been studied. The only example of homogeneous Einstein four-manifold which is curvature-Ricci commuting but not semisymmetric has been presented. Non-trivial examples of semi-symmetric homogeneous four-manifolds which are not locally symmetric, also Jacobi–Jacobi commuting manifolds which are not flat have been presented.
