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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.

Thank you in advance

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  • $\begingroup$ Didn't you already ask this question? mathoverflow.net/questions/229958/… $\endgroup$ – Nate Eldredge Apr 20 '16 at 18:54
  • $\begingroup$ Yes, but I can not do it, and want to know the case of the quaternionic Heisenberg group $\endgroup$ – Z. Alfata Apr 20 '16 at 19:10
  • $\begingroup$ Are you can detail your answer or can you please direct me to a reference ? $\endgroup$ – Z. Alfata Apr 20 '16 at 20:46

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