We introduce a new set of generators for the Chow ring of a matroid.  Geometrically, these generators behave like base-point-free divisors on a projective variety, and combinatorially, they encode matroid operations called principal truncations.  By combining the theory of Lorentzian (a.k.a. strongly log-concave) polynomials with the properties of these generators, we obtain a simplified proof of the degree one Hodge-Riemann relations for matroids, which is the combinatorially relevant portion of the Hodge theory of matroids by Adiprasito-Huh-Katz.