In several papers the authors and some of their collaborators published already a series of results concerning pseudo-Riemannian manifolds satisfying some “pseudo-symmetry” curvature condition. In this paper they continue this research. Let $(M, g)$ be a pseudo-Riemannian manifold, $R$ its curvature tensor and $C$ the corresponding Weyl tensor. They study manifolds $(M,g)$ such that $R\cdot C-C\cdot R$ and the tensor $Q(g,R)$, which they defined in earlier papers, are linearly dependent at any point of $M$. Here $R$ and $C$ act as derivations. Their main result is that such manifolds must be semi-symmetric, i.e. $R\cdot R= 0$, a condition which provided the starting point of their research on this type of conditions. Furthermore, they provide some examples of semi-symmetric warped products which satisfy the relation mentioned above and which illustrate their search for a possible inverse of their main result.