Straightforward proof of closedness

The formula for the basic form is:

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:

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