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∗φ=0$d*\varphi =0$, φ∧dφ=0$\varphi \wedge d\varphi =0$) 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.