A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

We answer in the affirmative question posed by Conti and Rossi [7,8] on existence of nilpotent Lie algebras dimension 7 with an Einstein pseudo-metric nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ signature $(3, 4)$ group 7, such that is not Ricci-flat. show cannot be induced any closed $G_2^*$-structure group. Moreover, some results harmonic $G_2^*$-structures arbitrary 7-manifold $M$ are given. In particular, prove underlying necessarily Einstein, but if it then