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We consider the group of vertical diffeomorphisms of a principal bundle whichare not automorphisms, and the corresponding group of maps with value in thestructure group. The associated Lie algebra of general vertical vector fieldshas a Lie bracket that is but an instance (in degree 0) of theFr\"olicher-Nijenhuis bracket for vector-valued differential forms. The inducesbracket on the maps with value in the Lie algebra of the structure grouphappens to reproduces an "extended" bracket introduced heuristically in physicsto study asymptotic symmetries of gravity. We thus correct the misconceptionthat the latter is a Lie algebroid bracket. We detail the elementary underlyinggeometry, whose exposition is absent from most standard treatments of bundlegeometry, and we discuss its relevance to gauge field theory.

Note on the group of vertical diffeomorphisms of a principal bundle, and its relation to the Fr\"olicher-Nijenhuis bracket