I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community.

(I already asked this question in Math stackexchange https://math.stackexchange.com/questions/4122536/reference-for-mathematical-palatini-formalism-of-general-relativity)

I am looking for a reference (book, review or research paper) of the Palatini or Tetradic Formulation of General Relativity from a more mathematical point of view. Up to now, I was only able to find some short physical introductions. So I would like to know if someone knows some more extensive discussion, maybe also from a mathematical point of view, i.e. by discussing frame bundles and the (mathematical) gauge theory point of view.