 If formula_22 is the frame bundle of a vector bundle formula_23, then formula_24 has fibre the general linear group formula_25 (either real or complex, depending on formula_26) where formula_27.This structure group has Lie algebra consisting of all formula_28 matrices formula_29, and these can be thought of as the endomorphisms of the vector bundle formula_26.Indeed there is a natural isomorphism formula_31.