Variation of quadratic Riemann Curvature tensor

[Qatawna blitz]

TL;DR Summary

How can I variate the quadratic Riemann curvature tensor, I tried raising and lowering the indices.

Hi,
Can you help me with this variation, I tried raising and lowering the indices.

I tried for months every possible method to reach the following answer without success.

 

[haushofer]

Please provide the reference if you want people to help. Your Langrangian suddenly contains two indices u and v, and my crystal ball provides no information about what that means.

 

[Qatawna blitz]

I think the two indicies uv in the answered lagrangian means variation with respect to g_uv
It's in the textbook: Introducing Einstein's General Relativity "Ray d'Inverno" @haushofer

 

[Mike_bb]

Hi.

$L_{uv} = \sqrt{-g}[g_{ud}R^{abcd}R_{abcv} + g_{vd}R^{abcd}R_{abcu} - \frac {1}{2}g_{uv}R^{abcd}R_{abcd}]$

I could be wrong but it looks like following expression:

$L_{uv} = \sqrt{-g}[2R_{uv} - \frac {1}{2}g_{uv}R]$