  We investigate 3-nondegenerate CR structures in the lowest possible dimension
7, and one of our goals is to prove Beloshapka's conjecture on the symmetry
dimension bound for hypersurfaces in $\mathbb{C}^4$. We claim that 8 is the
maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7,
which is achieved on the homogeneous model. This part (I) is devoted to the
homogeneous case: we prove that the model is locally the only homogeneous
3-nondegenerate CR structure in dimension 7.
