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Abstract

Since the mid-nineties it has gradually become understood that the Cartan prolongation of rank 2 distributions is a key operation leading locally, when applied many times, to all so-called Goursat distributions. That is those, whose derived flag of consecutive Lie squares is a 1-flag (growing in ranks always by 1). We first observe that successive generalized Cartan prolongations (gCp) of rank $k{+}1$ distributions lead locally to all special $k$-flags: rank $k{+}1$ distributions $D$ with the derived flag $\cal F$ being a $k$-flag possessing a corank 1 involutive subflag preserving the Lie square of $\cal F$. (Note that 1-flags are always special.) Secondly, we show that special $k$-flags are effectively nilpotentizable (or: weakly nilpotent) in the sense that local polynomial pseudo-normal forms for such $D$ resulting naturally from sequences of gCp's give local nilpotent bases for $D$. Moreover, the nilpotency orders of the generated real Lie algebras can be explicitly computed by means of simple linear algebra (for $k = 1$ this was done earlier in \cite{M2}, \cite{M5}). For $k = 2$ we also transform our linear algebra formulas into recursive ones that resemble a bit Jean's formulas \cite{J} for nonholonomy degrees of Goursat germs. Additionally it is shown that, when all parameters appearing in a local form for a special $k$-flag vanish, then such a distribution germ is also strongly nilpotent in the sense of \cite{AG} and \cite{M2}.