Can someone point me some reference for the moving frame theory in semi-Riemannian manifolds, using differential forms? In special, I'm looking for a version of Cartan's structural equations. I've already looked the references given in this answer, but it is not exactly what I'm looking for.
I also took a look in O'Neill's "The Geometry of Kerr Black Holes", but it wasn't helpful to me: the way he defines the connection forms $\omega_{ij}$ is different than the way he defines them in his first book Elementary Differential Geometry (indices swapped, and something more, it seems). I managed to adapt the connection equations, and his expression for the connection forms $\omega = dA \cdot A^t$ (page $91$), with some changes, but I'm failing to get the structural equations (page $95$). My level of background is more or less of this book.
I'm not familiar with tensors. Just more or less with differential forms.
If someone knows any text who does something like that, it would help me a lot. Thanks.
