Carnot Algebras and Sub-Riemannian Structures with Growth Vector (2,$\,$3,$\,$5,$\,$6)

We describe all Carnot algebras with growth vector $(2,3,5,6)$, their normal forms, an invariant that distinguishes them, and a basis change that reduces such an algebra to a normal form. For every normal form, we calculate the Casimir functions and symplectic foliations on the Lie coalgebra. We describe the invariant and the normal forms of left-invariant $(2,3,5,6)$-distributions. We also obtain a classification of all left-invariant sub-Riemannian structures on $(2,3,5,6)$-Carnot groups up to isometry and present models of these structures.