  Based on a fact that complex Clifford algebras of even dimension are
isomorphic to the matrix ones, we consider bundles in Clifford algebras whose
structure group is a general linear group acting on a Clifford algebra by left
multiplications, but not a group of its automorphisms. It is essential that
such a Clifford algebra bundle contains spinor subbundles, and that it can be
associated to a tangent bundle over a smooth manifold. This is just the case of
gravitation theory. However, different these bundles need not be isomorphic. To
characterize all of them, we follow the technique of composite bundles. In
gravitation theory, this technique enables us to describe different types of
spinor fields in the presence of general linear connections and under general
covariant transformations.
