 Heisenberg horizontal distribution

The horizontal smooth distribution on the real Heisenberg group is spanned by $E_1={\partial }_x$ and $E_2={\partial }_y+x{\partial }_z$. It is not an involutive distribution because their bracket is ${\partial }_z$, but their brackets span the full tangent space. A horizontal curve satisfies $\dot{x}=\alpha$, $\dot{y}=\beta$, and $\dot{z}=x\beta$.