Geodesic
Carnot Algebras and Sub-Riemannian Structures with Growth Vector (2,$\,$3,$\,$5,$\,$6)
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 237-246

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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.
Keywords: sub-Riemannian geometry, Carnot algebras, Carnot groups, left-invariant sub-Riemannian structures. 
@article{TM_2021_315_a16,
     author = {Yu. L. Sachkov and E. F. Sachkova},
     title = {Carnot {Algebras} and {Sub-Riemannian} {Structures} with {Growth} {Vector} (2,$\,$3,$\,$5,$\,$6)},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {237--246},
     publisher = {mathdoc},
     volume = {315},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_315_a16/}
}
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Yu. L. Sachkov; E. F. Sachkova. Carnot Algebras and Sub-Riemannian Structures with Growth Vector (2,$\,$3,$\,$5,$\,$6). Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 237-246. http://geodesic.mathdoc.fr/item/TM_2021_315_a16/