Connection { Free Differential

Both vector 1-forms (tensor elds of type (1,1)) and Lie brackets map (tangent) vectors to the same. The former do so in a manner that is linear over the ring of functions; the latter are bilinear, but only over the constants. By composing one Lie bracket and m vector 1-forms (we consider 2 m 4) in all possible ways many maps of vector elds are obtained, bilinear over the constants. We study linear combinations of these maps with some special properties: 1. Bilinearity over the functions. This yields the F-N bracket as a diierential invariant of pairs of vector 1-forms, as well as an additional diierential invariant J of triples of vector 1-forms; both are tensors of type (1,2). 2. Linearity over the functions in one variable, while the dependence on the other variable extends to an action on all tensor elds, as a derivation of type (0,1). We give a complete analysis of all linear relations between these diierential in-variants. Most derive from a known relation between F-N brackets, and from the Leibniz rule for derivations. There is, however, an additional relation for quadruples of vector 1-forms that is independent of these. The commutator of two derivations of type (0,1) gives rise to another, of type (0,2). Its \index" (i.e., its action on functions) is expressed in terms of F-N brackets and the J-invariant. Several of the calculations are made simple by studying an appropriate graph. Others are performed with the use of a computer.