  A study is made of left-invariant $\mathrm{G}_2$-structures with an exact
3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a
codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie
group $G$ cannot admit a left-invariant closed $\mathrm{G}_2$-eigenform for the
Laplacian and that any compact solvmanifold $\Gamma\backslash G$ arising from
$G$ does not admit an (invariant) exact $\mathrm{G}_2$-structure. We also
classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one
ideal equal to the complex Heisenberg Lie algebra which admit exact
$\mathrm{G}_2$-structures with or without special torsion. To achieve these
goals, we first determine the six-dimensional nilpotent Lie algebras
$\mathfrak{h}$ admitting an exact $\mathrm{SL}(3,\mathbb{C})$-structure $\rho$
or a half-flat $\mathrm{SU}(3)$-structure $(\omega,\rho)$ with exact $\rho$,
respectively.

Differential Geometry