# Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3$ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant vector fields on $H$.

And what has been the same for the sub-Laplacian on the quaternionic Heisenberg group $H^7 = \mathbb H \times \mathbb R^3$, where $\mathbb H$ is the space of the quaternion.