The Atiyah Lie algebroid associated to a -principal bundle over is a Lie algebroid structure on the vector bundle , the quotient of the tangent bundle of the total space by the canonical induced -action.
the Atiyah Lie algebroid sequence of is a sequence of Lie algebroids
is the Atiyah Lie algebroid
is the tangent Lie algebroid of .
The Lie bracket on the sections of is that inherited from the tangent Lie algebroid of .
To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of vector bundles. In that case one finds the curvature of the connection precisely as the obstruction to having a splitting even in Lie algebroids.
One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely -algebroids, in terms of an horizontal categorification of nonabelian Lie algebra cohomology: