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

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.