  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, hence, 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.
