I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 Lorentz (obeying the [1, -1, -1, -1] metric) index and 2 spinor indices (obeying the flat metric [1, 1, 1, 1]).

The problem is that such a tensor seems to be ill-defined in book definitions, I was unable to identify it as a tensor product of elements of the tangent space and cotangent space of some manifold.

Separating the Lorentz and spinor indices, it is clear that the Lorentz index stands for a 1-rank tensor (i.e. vector or one-form, depending on raise/lower index) of the spacetime manifold, while the spinor indices stand for a 2-rank tensor of a flat manifold (or a Clifford algebra?).

I also know that the Lorentz and spinor indices are two representations of the Lorentz group, but still, how to characterize that in terms of manifolds and differential geometry?