Alexei Kotov and Thomas Strobl have introduced a covariantized formulation of Yang–Mills–Higgs gauge theories whose main motivation was to replace the Lie algebra with Lie algebroids. This allows the introduction of a possibly non-flat connection \nabla on this bundle, after also introducing an additional 2-form \zeta in the field strength. We will study this theory in the simplified situation of Lie algebra bundles, i.e. only massless gauge bosons, and we will provide a physical motivation of \zeta. Moreover, we classify \nabla using the gauge invariance, resulting into that \nabla needs to be a Lie derivation law covering a pairing \Xi, as introduced by Mackenzie. There is also a field redefinition, keeping the physics invariant, but possibly changing \zeta and the curvature of \nabla. We are going to study whether this can lead to a classical theory, and we will realize that this has a strong correspondence to Mackenzie’s study about extending Lie algebroids with Lie algebra bundles. We show that Mackenzie’s obstruction class is also an obstruction for having non-flat connections which are not related to a flat connection using the field redefinitions. This class is related to \mathrm{d}^\nabla \zeta, a tensor which also measures the failure of the Bianchi identity of the field strength and which is invariant under the field redefinition. This tensor will also provide hints about whether \zeta can vanish after a field redefinition.
