  A nilpotent Lie algebra ${\mathfrak g}$ is said to be naturally graded if it
is isomorphic to its associated graded Lie algebra ${\rm gr} \mathfrak{g}$ with
respect to filtration by ideals of the lower central series. This concept is
equivalent to the concept of the Carnot algebra arising in sub-Riemannian
geometry and the geometric control theory.
  We classify finite-dimensional and infinite-dimensional naturally graded Lie
algebras (Carnot algebras) ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak
g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak
g}_{i{+}1}, \; \dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i
\ge 1. $$ For growth functions of such Lie algebras, we have the estimate $F(n)
\le \frac{3}{2}n{+}1$.
