  We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal
bundles, from the viewpoint of Lie algebraic approach to space. First we show
that if the Lie algebras of smooth sections of two Atiyah algebroids are
isomorphic, then the corresponding base manifolds are necessarily
diffeomorphic. Further, we give two characterizations of the isomorphisms of
the Lie algebras of sections for Atiyah algebroids associated to principle
bundles with semisimple structure groups. For instance we prove that in the
semisimple case the Lie algebras of sections are isomorphic if and only if the
corresponding Lie algebroids are, or, as well, if and only if the integrating
principal bundles are locally diffeomorphic. Finally, we apply these results to
describe the isomorphisms of sections in the case of reductive structure groups
-- surprisingly enough they are no longer determined by vector bundle
isomorphisms and involve divergences on the base manifolds.
