Tetrads and metrics I'm looking to do a similar calculation as in: Curvature tensor of 2-sphere using exterior differential forms (tetrads) in terms of tetrads/Vierbeins etc but I can't find any literature on _how_ one explicitly calculates the tetrads or even what they are. Any help or clarity would be greatly appreciated. Even if you could explain how to calculate this tetrad basis $e^\theta = rd\theta$ and $e^\phi = r \sin{\theta} d\phi$ in this case. Any references on the Riemann curvature tensor in terms of tetrads and the torsion tensor would be greatly appreciated.

I believe this is covered extremely well in John Stewart's Advanced General Relativity.%20Spinors.%20In%20Advanced%20General%20Relativity%20\(Cambridge%20Monographs%20on%20Mathematical%20Physics,%20pp.%2062-112\).%20Cambridge:%20Cambridge%20University%20Press.%20doi:10.1017/CBO9780511608179.003). It is a short textbook but extremely rich in content, certainly on the tetrad representation moving to Newman-Penrose formalism.

Alternatively you could look in the introduction / Chapter 1 of Chandrasekhar's Mathematical Theory of Black Holes.