Contracting the differential Bianchi identity R α α β β [ γ γ δ δ ; ε ε ] = 0 with g gives, using the fact that the metric tensor is covariantly constant, i.e. g ;γ = 0, R γ γ β β γ γ δ δ ; ε ε + R γ γ β β ε ε γ γ ; δ δ + R γ γ β β δ δ ε ε ; γ γ = 0