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The Spinor bundle is a vector bundle, but not part of the Tensor bundle.   

ITYM the ""Definition"" box of Tensor#As multidimensional arrays in this article. 

I've never seen a definition of the tensor bundle that included a structure group, although such a definition is customary for the more general Fiber bundle. Note that Spinor bundle refers to the Orthonormal frame bundle rather than to the Tangent bundle.   "