I am currently reading some lecture notes on Ricci flow and am not sure how the following identity is derived:
$\frac{\partial}{\partial t} \bigg( g^{ij}g^{kl}g^{ab}g^{cd}R_{ikac}R_{jlbd} \bigg) = 2\bigg(R^{ij}g^{kl}g^{ab}g^{cd} + g^{ij}R^{kl}g^{ab}g^{cd}+g^{ij}g^{kl}R^{ab}g^{cd} +g^{ij}g^{kl}g^{ab}R^{cd}\bigg)R_{ikac}R_{jlbd} \:+\:2<Rm,\frac{\partial}{\partial t}Rm> $
I know there are identities for the time derivative of the Riemann tensor in general, but I don't quite follow how the author has gone from the time derivative to the right-hand side, could someone explain?
Apply the product rule repeatedly and use the fact that $\frac{\partial}{\partial t}g^{ij}=2R^{ij}$. The final term arises from the fact that the metric is a symmetric form.