The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positivesemidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly containsthe class of matroids representable over the complex numbers. This connection was used by the first author toconstruct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a nonrepresentable hyperbolic matroid. The Vamos matroid and a generalization of it are to this day the only known ´instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids arenon-representable hyperbolic matroids by exploiting a connection, due to Jordan, between Euclidean Jordan algebrasand projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniformhypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraicversions of the generalized Lax conjecture.