I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann connections on the (orthonormal) frame bundle in a rigorous manner, rather than Koszul connections on the tangent bundle, and develops calculus directly on the frame bundle, rather than on spacetime itself.

Basically, I am interested in this, for the sake of being interested in it, however I have hopes that I might be able to use this formalism to attack some problems in my research.

To clarify a bit more:

- I am not looking for local tetrad formalism, however I am hoping that if what I am asking for exists, it will resemble local tetrad formalism a lot, but with globally defined quantities, as opposed to just local ones.
- Although I'll take any materials on the subject gladly, I really would prefer if the resource treated lagrangian formalism. I am mostly aware of how Ehresmann connections on the frame bundle work, but I have absolutely no clue how to do langrangian formalism with it, and this is an absolute necessity for my work.
- I have Kobayashi & Nomizu for necessary extra mathematical details, but I'd prefer this resource to be generally self-contained.