Solving the equation for
Let us investigate the equation for :
Let us solve it perturbatively as a series in . We have:
 

We can proceed similarly to higher orders. An immediate consistency check on Eq. (15) is
that is closed:
This should be checked order by order. And then we should prove order by order that is exact.
The validity of Eq. (17) can be proven by induction order by order as follows:
The first term is zero by Jacobi identity. That the second term
is zero can be proven using an additional assumption that is
invariant; this should be checked order by order.
Then we should check that is exact.
Suppose that all these assumptions are true. Then we have a “formal solution”: