  The Poincare'-Dulac normal form of a given resonant system is in general non
unique; given a specific normal form, one would like to further reduce it to a
simplest normal form. In this note we give an algorithm, based on the Lie
algebraic structure of the set of normal forms, to obtain this. The algorithm
can be applied under some condition, non generic but often met in applications;
when applicable, it only requires to solve linear equations, and is more
powerful than the one proposed in previous work by the same author [Lett. Math.
Phys. 42, 103-114; and Ann. I.H.P. 70, 461-514].
