In the paper
Cao, Huai-Dong, Geometry of Ricci solitons, Chin. Ann. Math., Ser. B 27, No. 2, 121-142 (2006). ZBL1102.53025, there is a statement saying that; if $g_{ij}$ is a complete gradient Ricci soliton on the Riemannian manifold $M$, i.e., $$R_{ij}+\nabla_i\nabla_jf=\lambda g_{ij},$$ then we obtain $$\nabla_iR_{jk}-\nabla_jR_{ik}+R_{ijkl}\nabla_lf=0.$$ I can not get how this result comes. Please help me to find the intermediate calculation.