A coherent approach to the description of double excitations in correlated materials is presented: We derive stringent mathematical conditions on the algebraical structure of the Bethe-Salpeter and time-dependent density functional theory kernels that avoid the occurrence of spurious and nonphysical excitations. We discuss how these conditions need to be respected at any level of approximation, including the commonly used local density and static screening approximations. We propose a correlated kernel for the Bethe-Salpeter equation, and we illustrate several aspects of our approach with numerical calculations for model molecular systems.