  The group of vertical diffeomorphisms of a principal bundle forms the
generalised action Lie groupoid associated to the bundle. The former is
generated by the group of maps with value in the structure group, which is also
the group of bisections of the groupoid. The corresponding Lie algebra of
general vertical vector fields is generated by maps with value in the Lie
algebra of the structure group. The bracket on these maps, induces by the
bracket of vertical vector fields, is an ``extended" bracket on gauge
parameters: it has been introduced heuristically in physics, notably in the
study asymptotic symmetries of gravity. Seeing the set of Lie algebra-valued
maps as sections of the action Lie algebroid associated to the bundle, the
extended bracket is understood to be a Lie algebroid bracket on those sections.
Here, we highlight that this bracket can also be seen to arise from the
Fr\"olicher-Nijenhuis bracket of vector-valued differential forms. The benefit
of this viewpoint is to insert this extended bracket within the general
framework of derivations of forms on the bundle. Identities relating it to
usual operations -- inner product, exterior and (Nijenhuis-) Lie derivative --
are immediately read as special cases of general results. We also look at the
generalised gauge transformations induced by vertical diffeomorphisms, and
discuss their peculiar features. In particular, locally, and contrary to
standard gauge transformations arising from vertical bundle automorphisms, they
are distinguishable from local gluings when iterated. Yet, the gauge principle
still holds.
