Nilmanifolds with a calibrated G2-structure

We introduce obstructions to the existence of a calibrated G2-structure on a Lie algebra gg of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from gg to a six-dimensional Lie algebra hh with kernel contained in the center of gg, then hh has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G2-structure.