  In the main part of this paper a connection is just a fiber projection onto a
(not necessarily integrable) distribution or sub vector bundle of the tangent
bundle. Here curvature is computed via the Froelicher-Nijenhuis bracket, and it
is complemented by cocurvature and the Bianchi identity still holds. In this
situation we determine the graded Lie algebra of all graded derivations over
the horizontal projection of a connection and we determine their commutation
relations. Finally, for a principal connection on a principal bundle and the
induced connection on an associated bundle we show how one may pass from one to
the other. The final results relate derivations on vector bundle valued forms
and derivations over the horizontal projection of the algebra of forms on the
principal bundle with values in the standard vector space.
