Purely coclosed G2-structures on nilmanifolds

We classify seven-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed G₂-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 φ which is a pure coclosed G₂-structure (i.e., it satisfies $d\ast \phi =0$, $\phi \wedge d\phi =0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed G₂-structure for the rest of them.