How Does Tetrad Formalism Relate to the Hilbert-Einstein Action?

[Andre' Quanta]

I would like to write the HIlbert-Einstein action using tetrad formalism.
I have seen in a paper that the lagrangian is the contraction between a levi civita tensor, against two veilbeins and a two form curvature.
The problem is that i can' t prove that this is the same as the scalar curvature moltiplied by the determinant of the metric.
Could you help me? I am sure of the result, but i can' prove it

 

[WannabeNewton]

See section 3.2: "Covariant Loop Quantum Gravity", by Rovelli and Vidotto, published by Cambridge University Press.

 

[Andre' Quanta]

See section 3.2: "Covariant Loop Quantum Gravity", by Rovelli and Vidotto, published by Cambridge University Press.

Thanks you really much :)
I had a launch with Carlo Rovelli and Francesca Vidotto, they were really nice: it s a surprise to see the solution in their book