I'm reading about the Ricci tensor, and I've found the following statement that is given without proof:
For a point $p$ on a Riemannian manifold, and coordinate vector fields $X_{\alpha}$, $X_{\beta}$, and $X_{\gamma}$ defined in a neighborhood about $p$, $R(X_{\alpha},X_{\gamma})X_{\beta} + R(X_{\beta},X_{\gamma})X_{\alpha} = -3(\nabla^2_{\alpha,\beta}X_{\gamma} + \nabla^2_{\beta,\alpha}X_{\gamma})$
Using the fact that $R(X_{\alpha},X_{\gamma})X_{\beta} = \nabla^2_{\alpha,\gamma}X_{\beta} - \nabla^2_{\gamma,\alpha}X_{\beta}$ and the Bianchi identity, I have been able to show that $R(X_{\alpha},X_{\gamma})X_{\beta} - R(X_{\beta},X_{\gamma})X_{\alpha} = \nabla^2_{\alpha,\beta}X_{\gamma} - \nabla^2_{\beta,\alpha}X_{\gamma}$, however is not quite what I was trying to prove. I would greatly appreciate any help.