Automorphisms of non-singular nilpotent Lie algebras

For a real, non-singular, 2-step nilpotent Lie algebra $\mathfrak{n}$, the group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n})$, where$\Aut_0(\mathfrak{n})$is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the 1-dimensional group of dilations. Maximality of some automorphisms groups of$\mathfrak{n}$follows and is related to how close is$\mathfrak{n}$to being of Heisenberg type. For example, at least when the dimension of the center is two,$\dim \Aut(\mathfrak{n})$is maximal if and only if$\mathfrak{n}$is type$H$. The connection with fat distributions is discussed.