does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds. It is said that $$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$ because we are in dimension 4, where$\check{R}_{ab}=R_{ajkl}R^{jkl}_b$ and $\mathring{W}$ is Weyl acting on symmetric tensor. is there is a simple idea to get it, except to devellop everything? Infact I am trying to get it from a moving frame point of view, and more generally, I am looking for any reference making computation about curvature functionnal, Einstein metic and Bach tensor from the moving frame point of view. Thanks in advance

Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that $$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$ because we are in dimension 4, where$\check{R}_{ab}=R_{ajkl}R^{jkl}_b$ and $\mathring{W}$ is Weyl acting on symmetric tensor. Is there is a simple idea to get it, except to develop everything? In fact I am trying to get it from a moving frame point of view, and more generally, I am looking for any reference making computation about curvature functional, Einstein metric and Bach tensor from the moving frame point of view. Thanks in advance.