This is probably a silly question but please humor me. In an orthonormal tetrad formulation of GR we define (co)frame fields $e^{a},e_{a}$ that everywhere obeys $e^{a}e_{b}=\delta_{b}^{a}$.
Choosing a torsion free Levi-Cevita -esque connection defined by:
$$de^{a}=\omega_{b}^{a}e^{b}$$
we can solve for our spin connection $\omega^{ab}$ (where all coordinate indices have been suppressed). My question is pretty straight forward: Is the spin connection $\mathfrak{so}(1,3)$ or $\mathfrak{spin}(1,3)$ valued? Now before someone points it out...Yes I understand that they're isomorphic as Lie algebras; however the spin connection is utilized in both transformations of mixed indice tensors (having Lorentzian as well as coordinate indices) and also spinors.
So when we're looking at sections of frame bundle surely the same object can't be describing transformations of sections of two different frame bundles? Could someone please elaborate here? Maybe it's really simple and I'm just tired lol. I apologize, if I have taken some factors for granted (physics)
Note: there is a factor of 1/2 that may explain it somewhat when I think about the exponential map...