  We classify real and complex infinite-dimensional narrow positively graded
Lie algebras ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with
properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \;
\dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i \ge 1. $$ In the
proof of the main theorem we apply successive central extensions of
finite-dimensional Carnot algebras. In sub-Riemannian geometry, control theory,
and geometric group theory, Carnot algebras play a significant role.
