Purely coclosed G(2)-structures on nilmanifolds

We classify seven-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed G(2)-structures. This is done by going through the list of all seven-dimensional nilpotent Lie algebras given by Gong, providing an example of a left-invariant 3-form phi which is a pure coclosed G(2)-structure (i.e., it satisfies d*phi=0$d*\varphi =0$, phi perpendicular to d phi=0$\varphi \wedge d\varphi =0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed G(2)-structure for the rest of them.