I'm reading "Textbook in Tensor Calculus and Differential Geometry" by Prasun Kumar Nayak and came across the Weyl tensor/projective curvature tensor $C_{kijl}$. The book states that $$C_{kijl}=R_{kijl}+\frac{1}{1-N}(g_{kj}R_{il}-g_{kl}R_{ij}) \tag{1}$$
However I found on Wikipedia that $$C_{iklm}=R_{iklm}+\frac{1}{N-2}(R_{im}g_{kl}-R_{il}g_{km}+R_{kl}g_{im}-R_{km}g_{il})+\frac{1}{(N-1)(N-2)}R(g_{il}g_{km}-g_{im}g_{kl}) \tag{2}$$
It was not immediately obvious to me that both of these are equivalent to each other?
The first equation $C_{kijl}=R_{kijl}+\frac{1}{1-N}(g_{kj}R_{il}-g_{kl}R_{ij})$ is not the usual Weyl tensor. It is Weyl projective curvature tensor and it is different from Weyl tensor. It usually denoted by $W_2$. See definition 2.1 here.
In the NAYAK's book you should read "projective curvature tensor or Weyl tensor" as "projective curvature tensor or projective Weyl tensor".