  This work extends previous developments carried out by some of the authors on
Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan
connections in a framework relying on two Atiyah Lie algebroids based on a
$H$-principal fiber bundle $\mathcal{P}$ and its associated $G$-principal fiber
bundle $\mathcal{Q} := \mathcal{P} \times_H G$, where $H \subset G$ defines the
model for a Cartan geometry. The first main result of this study is a
commutative and exact diagram relating these two Atiyah Lie algebroids, which
allows to completely characterize Cartan connections on $\mathcal{P}$.
Furthermore, in the context of gravity and mixed anomalies, our construction
answers a long standing mathematical question about the correct
geometrico-algebraic setting in which to combine inner gauge transformations
and infinitesimal diffeomorphisms.
