On the finiteness of differential invariants

In this paper we show how Weil's theory of near points yields a new light on the classical approaches to the study of the differential invariants of a sheaf of tangent vector fields. We give conditions for the existence of invariant derivations for a sheaf of tangent vector fields, which allows to apply Lie's algorithm to obtain new differential invariants as quotients of Jacobian determinants of known ones. We give sufficient conditions for the asymptotic stability of the symbol of a sheaf of tangent vector fields and prove our main result, a finiteness theorem for the differential invariants of a sheaf of Lie algebras which simplifies and improves on the treatment given in J. Differential Geom. 10 (1975) 249-416.