 The adjoint bundle of "P" is the associated bundle
The adjoint bundle is also commonly denoted by formula_4.Explicitly, elements of the adjoint bundle are equivalence classes of pairs ["p", "x"] for "p" ∈ "P" and "x" ∈ formula_1 such that
for all "g" ∈ "G".Since the structure group of the adjoint bundle consists of Lie algebra automorphisms, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over "M".