 Steady gradient Ricci soliton scalar identity

For $\mathrm{Ric}+{\nabla }^{2}f=0$, the contracted Bianchi identity and the commutation identity $\Delta \nabla f=\nabla \Delta f+\mathrm{Ric}(\nabla f)$ imply $\nabla R=2\mathrm{Ric}(\nabla f)$. Meanwhile $\nabla |\nabla f{|}^2=-2\mathrm{Ric}(\nabla f)$, so their sum is constant on every connected component.