  We classify 7-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 7-dimensional nilpotent Lie
algebras given by Gong, providing an example of a left-invariant 3-form
$\varphi$ which is a pure coclosed $G_2$-structure (that is, it satisfies
$d*\varphi=0$, $\varphi \wedge d\varphi=0$) 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.
