Coclosed $G_2$-structures inducing nilsolitons

We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $\mathfrak{g}$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $\mathfrak{g}$ to a six-dimensional Lie algebra $\mathfrak{h}$, with kernel contained in the center of $\mathfrak{g}$, then any coclosed $G_2$-structure on $\mathfrak{g}$ induces a closed and stable three form on $\mathfrak{h}$ that defines an almost complex structure on $\mathfrak{h}$. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed $G_2$-structures. We also prove that each one of these Lie algebras has a coclosed $G_2$-structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed $G_2$-structures. The existence of contact metric structures is also studied.