Suppose we have a Gauss-Bonnet like theory which its Lagrangian can be written as,

\[\mathcal{L}=aR^2+bR_{\mu\nu}R^{\mu\nu}+cR_{\mu\nu\lambda\sigma} R^{\mu\nu\lambda\sigma}\]

I want to write above after the ADM decomposition. I know that Riemann tensor can be written in term of the ADM variable via Gauss, Codazzi and Ricci equations. Namely via Eq 2.11 in this paper. Essentially what I am looking after is to write the above Lagrangian as 4.25 in the same paper. So far I tried to use the definitions of the terms appearing in 4.25 but I was not so successful. (The definitions are given in 2.13). Any one has any idea about this?