Following the previous note where we discuss about the norm of the tensor in terms of the trace-free Ricci tensor, i.e. the following identity

holds. Today, we derive some further identites involving the Ricci tensor . As we have already seen from the previous notes, the trace-free Ricci tensor is given by

First, we calculate . For simplicity, we denote by and $S$ the Ricci tensor and the scalar curvature respectively. By definition, we have