Construction of base form
Let denote the equivariant form, given by Eq. (30). We will denote the corresponding base form :

We will work under the assumption that:
  • the action of on does not have fixed points

Therefore can be considered a principal -bundle. In order to construct the base form from the Cartan’s , we first choose on this principal bundle some connection . (We understand the connection as a -valued 1-form on computing the “vertical component” of a vector.) Then we apply the horizonthal projection i.e. replace

Finally, we replace with the curvature of the connection ; we get:

To conclude:
Generally speaking, it is not true that descends from to . But we found a class of subalgebras of , of the form , such that for any subalgebra from this class we can construct an equivariant form . Then the standard procedure can be used to construct the corresponding base form which descends to . The result, generally speaking, does depend on the choice of .