Baillon–Haddad theorem ($⟨\nabla f(u)-\nabla f(v),u-v⟩\ge L^{-1}\Vert \nabla f(u)-\nabla f(v){\Vert }^2$)

On the whole Euclidean space, an $L$-Lipschitz gradient of a convex function is $1/L$-cocoercive. Subtract the supporting affine function at one point, apply the descent lemma to a gradient step at the other point, then interchange the points and add. The conclusion needs convexity but no second derivative.