# On $\mathrm{G}_2$ and Sub-Riemannian Model Spaces of Step and Rank Three

## TL;DR

This paper classifies all sub-Riemannian model spaces with step and rank three, revealing their structure, parameters, and connection to exceptional Lie algebras, and showing no free nilpotentization exists in this class.

## Contribution

It provides a complete classification of step and rank three sub-Riemannian model spaces, including their families, parameters, and realization of $rak{g}_2$ as isometry algebras.

## Key findings

- Three families of model spaces based on nilpotentization
- No nontrivial free nilpotentization for these spaces
- Realization of $rak{g}_2^c$ and $rak{g}_2^s$ as isometry algebras

## Abstract

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces. In particular, there are no nontrivial sub-Riemannian model spaces of step and rank three with free nilpotentization. We also realize both the compact real form $\mathfrak{g}_2^c$ and the split real form $\mathfrak{g}_2^s$ of the exceptional Lie algebra $\mathfrak{g}_2$ as isometry algebras of different model spaces.