  We answer in the affirmative the question posed by Conti and Rossi on the
existence of nilpotent Lie algebras of dimension 7 with an Einstein
pseudo-metric of nonzero scalar curvature. Indeed, we construct a
left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a
nilpotent Lie group of dimension 7, such that $g$ is Einstein and not
Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any
left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some
results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold
$M$ are given. In particular, we prove that the underlying pseudo-Riemannian
metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily
Einstein, but if it is Einstein then it is Ricci-flat.
