Suppose that the Lie superalgebra is realized as vector fields on some supermanifold . This means that pseudo-differential forms on are a -differential module (= representation of ). Consider the equation:

We can write such an equations in any -differential module , not necessarily PDFs on .

Now suppose that is a -differential module (= representation of ). Since is embedded into , we can still write the Cartan Eq. (15):

where we use the notations of Eqs. (7), (14).

Then, given any -closed form , consider the following anstaz for a solution of Eq. (15):

where is from Eq. (13). We will show that under certain conditions this substitution solves Eq. (15).