We report on dark-exciton dynamics in a single quantum dot embedded in a bimodal cavity under the effect of an external magnetic field and laser excitation. By means of a fast laser pulse in resonance with the cavity modes and a varying intensity magnetic field in different tilted angles, we are able to identify the optimal condition where dark-exciton states can be efficiently populated. Using the set of parameters, we focus on the steady regime for the system that is continuously excited. Our results show that the presence of dark-exciton states strongly modifies the cavity modes' occupation and, for a specific set of parameters, only dark states are present in the cavity.