to the “rolling balls problem”, treated by Cartan in [@Ca]. The prolongation, representing -roughly speaking- the maximal possible symmetry of the distribution, in the case of ${\mathfrak n }_{2,3}$ is the exceptional Lie algebra ${\mathfrak g }_2$ [@BM]. The Lie algebra ${\mathfrak n }_{3,2}$ was studied more recently in [@My] in the context of the geometric characterization of the so-called Maxwell set, wave fronts and caustics (see also [@MA]).

We complete the work with a study of the group of automorphisms of the Lie algebras ${\mathfrak n }_{2,3}$ and ${\mathfrak n }_{3,2}$. The corresponding structure is described and in particular the subgroup of orthogonal automorphisms is determined.

Following [@FS] and the considerations above all Lie algebras here are over a field $K$ of characteristic 0, nevertheless some results in Section 3 could be still true for fields of a characteristic different from 2.

Free and free metabelian nilpotent Lie algebras
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Let ${\mathfrak g }$ denote a Lie algebra. The so-called central descending and ascending series of ${\mathfrak g }$, respectively $\{C^r({\mathfrak g })\}$ and $\{C_r({\mathfrak g })\}$ for all $r\geq 0$, are constituted by the ideals in ${\mathfrak g }$, which for non-negative integers $r$, are given by $$\begin{array}{rclrcl}
