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

$R\xb7C-C\xb7R$ 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.

$R\xb7R=0$, 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.