I hope this is not a silly question but I am trying to understand how this part of the equation works:
$$ \nabla_{\lambda} \left( \nabla_{\mu}(R_{\nu \lambda}) + \nabla_{\nu}(R_{\mu \lambda}) \right) $$
Where $R_{xx}$ is the Ricci tensor.
My question: Am I required to work out the covariant derivatives separate then work out the 'outer' covariant derivative of the result or should I use the double covariant derivative rule twice like so:
$$ \begin{align} \nabla_a \nabla_b h_{cd} &= \partial_a ( \partial_b h_{cd} - \Gamma^f_{bc} h_{fd} - \Gamma^f_{bd} h_{cf} ) \\ &\qquad \qquad - \Gamma^e_{ab} ( \partial_e h_{cd} - \Gamma^f_{ec} h_{fd} - \Gamma^f_{ed} h_{cf} ) \\ &\qquad \qquad - \Gamma^e_{ac} ( \partial_b h_{ed} - \Gamma^f_{be} h_{fd} - \Gamma^f_{bd} h_{ef} ) \\ &\qquad \qquad - \Gamma^e_{ad} ( \partial_b h_{ce} - \Gamma^f_{bc} h_{fe} - \Gamma^f_{be} h_{cf} ) \end{align} $$