In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_1,\mathrm{\dots },X_m$ on ${\mathbb{R}}^N$, we ask whether there exists a homogeneous Carnot group $\mathbb{G}=(\mathbb{R}{}^N,\circ ,\delta {}_{\lambda })$ such that ${\sum }_i{X}_i^2$ is a sub-Laplacian on $\mathbb{G}$. We find necessary and sufficient conditions on the given vector fields in order to give a positive answer to the question. Moreover, we explicitly construct the group law i as above, providing direct proofs. Our main tool is a suitable version of the Campbell-Hausdorff formula. Finally, we exhibit several...