  We give a semi-small orthogonal decomposition of the Chow ring of a matroid
M. The decomposition is used to give simple proofs of Poincar\'e duality, the
hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring,
recovering the main result of [AHK18]. We also show that a similar semi-small
orthogonal decomposition holds for the augmented Chow ring of M.
