The Carnot sphere is the set:
$$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
Show the existence of a constant $c_1 > 0$ such that for all $(p,q,r) \in B(1)$, we have
$$c_1^{-1} \leq p^2 + q^2 + |r| \leq c_1$$