Purely coclosed ${\mathrm{G}}_2$-structures on nilmanifolds -- II

This paper completes the classification of seven-dimensional nilpotent Lie groups endowed with a left-invariant purely coclosed ${\text{G}}_2$-structure, initiated by the first-named author and collaborators. In this previous work, the authors provided the classification of decomposable seven-dimensional nilpotent Lie groups and of the indecomposable ones up to step $4$ of nilpotency. Here, we address the case of indecomposable $5$- and $6$-step nilpotent Lie groups.