  This paper derives new identities for the Weyl tensor on a gradient Ricci
soliton, particularly in dimension four. First, we prove a
Bochner-Weitzenb\"ock type formula for the norm of the self-dual Weyl tensor
and discuss its applications, including connections between geometry and
topology. In the second part, we are concerned with the interaction of
different components of Riemannian curvature and (gradient and Hessian of) the
soliton potential function. The Weyl tensor arises naturally in these
investigations. Applications here are rigidity results.
