Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) connection $\omega$. This lets us describe a Riemannian metric on $M$ using a vector bundle map

$e: TM\rightarrow T$

We have that

$\omega\in so(4)$ (it's a $SO(4)$ connection) and $e\in \mathbb{R}^4$.

We can introduce now a $SO(5)$ connection $A$ such that

$A=\omega+e$ $\hspace{1.0cm}$(because $so(5)=so(4)\oplus \mathbb{R}^4$)

**My question is**: if I introduce a spin structure (spin bundle), can I define the following Dirac operator:

$D=\gamma^\mu(\partial_\mu+A_\mu)$ ?

Can I consider $A$ as a "pure" (gauge) connection as in the standard case?