  To compute the unique formal normal form of families of vector fields with
nilpotent linear part, we choose a basis of the Lie algebra consisting of
orbits under the linear nilpotent. This creates a new problem: to find explicit
formulas for the structure constants in this new basis. These are well known in
the 2D case, and recently expressions were found for the 3D case by ad hoc
methods. The goal of the present paper is to formulate a systematic approach to
this calculation.
  We propose to do this using a rational method for the inversion of the
Clebsch-Gordan coefficients. We illustrate the method on a family of 3D vector
fields and compute the unique formal normal form for the Euler family both in
the 2D and 3D case.
