The present paper considers biaxial nematogenic lattice models, involving particles of D2h symmetry, whose centers of mass are associated with a three-dimensional simple-cubic lattice. The pair potential is isotropic in orientation space and restricted to nearest neighbors. Let two orthonormal triads define orientations of a pair of interacting particles. The investigated potential models are quadratic with respect to the nine scalar products between the two sets of unit vectors. Actually, based on available geometric identities, these expressions can be reduced to diagonal form containing only the scalar products between corresponding unit vectors and depending on three parameters. Over the years, this comparatively simple functional form has also proven to be rather versatile. By now, various sets of potential parameters capable of producing mesogenic behavior of some kind have been proposed and studied in the literature. A new and simplified form was recently proposed and investigated by Sonnet, Virga, Durand, and De Matteis [ A. M. Sonnet, E. G. Virga and G. E. Durand Phys. Rev. E 67 061701 (2003); G. De Matteis and E. G. Virga Phys. Rev. E 71 061703 (2005)] and is known to support a biaxial phase at sufficiently low temperature. Following the idea of the above authors, we have studied a more extended range of parameters, including cases where biaxiality cannot be sustained in the pair ground state. In cases where a biaxial phase survives, an appropriate mean-field analysis may predict the existence of a direct second-order transition to the isotropic phase as well as a second-order sequence isotropic-to-uniaxial-to-biaxial. A second-order phase transition is also predicted, which involves isotropic and uniaxial phases only. Monte Carlo simulations have been carried out as well, for a few points in the parameter space, and found to produce results which partly confirm mean-field predictions.