The author classifies left invariant Einstein metrics on some class of 7-dimensional solvable Lie groups using the variational approach, proposed by J. Lauret. The problem reduces to the classification of some solvable 7-dimensional Einstein metric Lie algebras , that is, a Lie algebra together with an Euclidean metric which satisfy the Einstein equation.
More precisely, let be a metric nilpotent Lie algebra and a -symmetric derivation of . Then the solvable metric Lie algebra , where is a semidirect sum and is a natural extension of the metric such that , is called a metric rank one solvable extension of . It is known that if is an Einstein metric Lie algebra which contains as a codimension one (metric) nilpotent ideal, then is a uniquely defined metric rank one solvable extension of .
The author classifies all metric rank one extensions of 6-dimensional metric nilpotent Lie algebras which are Einstein metric Lie algebras. She finds 34 metric nilpotent 6-dimensional Lie algebras, which have such extension.