I am studying the tetradic Palatini formalism of general relativity. In this formalism, one usually considers a manifold $M$, which is either non-compact or compact with Euler-characteristic $\chi(M)=0$ so that there exist Lorentzian metrics on $M$. Choosing such a metric $g$ on $M$, one defines the bundle of orthonormal co-frames on $M$, denotes $\mathcal{F}_{\mathrm{ort}}(TM)$. The two variables of interest in the Palatini formalism are the "frame field", which is a bundle isomorphism $e:\mathcal{T}\to TM$, where $\mathcal{T}:=\mathcal{F}_{\mathrm{ort}}(TM)\times_{\rho}\mathbb{R}^{1,3}$ denotes the associated vector bundle using the fundamental representation $\rho$ of $\mathrm{SO}(1,3)$ and the "Cartan connection" $\omega$, which is a connection $1$-form of $M$. Now, in many texts, $\omega$ is also referred to as "spin connection", a terminology which is usually also used for connections of a spinor bundle. So, a natural question is the following:

How is the Cartan connection appearing in the tetradic formulation of gravity related to a connection of a spinor bundle?