Semi-reducible pseudo-Riemannian spaces with additional conditions

We consider semi-reducible pseudo-Riemannian spaces with algebraic conditions on the Ricci tensor and the Riemann tensor. For almost Einstein and weakly recurrent spaces we find the type of tensor characteristic of semi-reducibility. Semi-reducible almost Einstein spaces and weakly recurrent spaces are divided into types depending on the properties of the vector fields that exist in them by necessity. The study is carried out locally in the tensor form.