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 ∇ on this bundle, after also introducing an additional 2-form ζ 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 ζ . Moreover, we classify ∇ using the gauge invariance, resulting into that ∇ needs to be a Lie derivation law covering a pairing Ξ , as introduced by Mackenzie. There is also a field redefinition, keeping the physics invariant, but possibly changing ζ and the curvature of ∇ . 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 d∇ζ , 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 ζ can vanish after a field redefinition.