  There is the notion of action Lie algebroids, containing information about
Lie algebras and their actions, which is why it is natural to generalise gauge
theories to a formulation using Lie algebroids; these allow structure functions
in general. This is for example done in the formulation of curved
Yang-Mills-Higgs gauge theory, formulated by Alexei Kotov and Thomas Strobl. We
will discuss how to formulate infinitesimal gauge transformations using Lie
algebroids in such a way that these close as algebra. For this we are going to
generalise infinitesimal gauge transformations of Yang-Mills-Higgs gauge
theories to derivations on vector bundle $V$-valued functionals. In the context
of gauge theory, we will motivate that those vector bundles $V$ will be the
pullback of another bundle $W$, and the gauge transformations as derivations
will be induced by a Lie algebroid connection on $W$, using a more general
notion of pullback connections. This also supports the usage of arbitrary types
of connections on $W$ in the definition of the infinitesimal gauge
transformation, not just canonical flat ones as in the classical formulation.
As usual, these derivations are parametrised and their parameters have to be
generalised to functionals, especially the parameters themselves can have a
non-trivial gauge transformation. We then discuss under which conditions this
algebra of derivations gives a closed algebra, also by providing a natural Lie
bracket on the space of parameters, and we are going to see that closure is
strongly related to the vanishing of a tensor known as the basic curvature.
