Narrow positively graded Lie algebras

We classify real and complex infinite-dimensional narrow positively graded Lie algebras $\mathfrak{g}={\oplus }_{i=1}^{+\mathrm{\infty }}{\mathfrak{g}}_i$ 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.