Fully attractive and partly repulsive molecular interactions, characteristic of biaxial liquid crystal molecules, are described by a class of quadratic Hamiltonians, originally put forward by Straley. The partly repulsive interactions require a minimax principle for the associated mean-field free energy. By analyzing the different phases in a bifurcation scenario, we show that the phase diagram described by Sonnet et al. (Phys. Rev. E 67 061701 (2003)) is universal. Our model predictions are in agreement with recent observations on both V-shaped and tetrapodal molecules.