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By means of a global Lifting method for homogeneous operators proved by Folland in [On the Rothschild-Stein lifting theorem, Comm. PDEs, 1977], there exists a Carnot group $\\mathbb{G}$ and a polynomial surjective map $\\pi:\\mathbb{G}\\to \\mathbb{R}^n$ such that $\\mathcal{L}$ is $\\pi$-related to a sub-Laplacian $\\mathcal{L}_{\\mathbb{G}}$ on $\\mathbb{G}$. 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