It was Gianna Stefani who first started to look for something simpler than canonical exponential coordinates of the 1st or 2nd kind - in her Bierutowice 1984 talk. Then she used that to locally simplify the control systems linear in controls - to define a prototype of the nilpotent approximation (NA in short) of the initial system. Agrachev & Sarychev joined in in 1987, Hermes & Kawski in 1991, Risler in 1992, Bellaiche in 1996. That last contibutor proposed an algorithm, of improving a given set of local coordinates to privileged (or: adapted) ones, that was purely polynomial, avoiding any exponentiation. In short (perhaps too short) Bellaiche successfully debugged Stefani's original approach of 1984. A little polished version of his procedure will be reproduced during the talk.

The second part is aimed at showing that the Bellaiche proposal is hardly a fully blown algorithm as in the title above. Since it is general, it is also cumbersome and - potentially - extremely memory-thirsty. It also leads sometimes to illisible visualisations of NA's. (A given NA has, as a rule, a plethora of various visualisations.)

In concrete classes of distributions dynamic modifications of `polynomial Bellaiche' are needed that would lead to much simpler visualisations. This is particularly important in the SR geometry when one reduces a local minimization problem to a simpler one showing up in the NA of an SR structure. Two instances of such `much simpler' visualisations will be given.