Straightforward proof of closedness
The formula for the basic form is:

(16)

This expression should be understood as follows. Notice that is an element of , i.e. an inhomogeneous differential form on with values in polynomial functions on . We then evaluate the function on ; for example the polynomial evaluates as follows:

We have to prove that defined in Eq. (16) is horizonthal and closed. The horizonthality follows immediately from the definition, because:
  • is essentially the projector on the horizonthal forms and

  • is automatically horizonthal

It remains to prove that is a closed form. We have:

(17)

where the following notations are assumed:
  • the symbol denotes the symmetrically ordered expression:

  • in the first line, the acts on everything that is (after symmetrization) to the right of it

in particular, it acts on , giving , which cancels with the second line