Trace-free differential invariants of triples of vector 1-forms

It is shown that the “trace-free” differential invariants of triples of vector 1-forms form a space of dimension 13. Twelve of these are accounted for by constructions based on the known bilinear “bracket” of vector 1-forms. We find one that is new, and exhibit it in various forms, including one that shows an unusual symmetry: it alternates in the three vector 1-forms and is a tensor of type (1,2), symmetric in its covariant part. Two-dimensional manifolds admit yet another new invariant.