On Einstein Extensions of Nilpotent Metric Lie Algebras

The main result of the article is as follows: If a nilpotent noncommutative metric Lie algebra $(\mathfrak n,Q)$ is such that the operator $\operatorname{Id}-\frac{\operatorname{trace}(\mathrm{Ric})}{\operatorname{trace}(\mathrm{Ric}^2)}\mathrm{Ric}$ is positive definite then every Einstein solvable extension of $(\mathfrak n,Q)$ is standard. We deduce several consequences of this assertion. In particular, we prove that all Einstein solvmanifolds of dimension at most 7 are standard.