Journal of Lie Theory

Consider a two-step nilpotent Lie algebra n with a special nice basis as introduced by Y. Nikolayevsky [Einstein solvmanifolds and the pre-Einstein derivation, Trans. Amer. Math. Soc. 363 (2011) 3935--3958] endowed with an inner product which makes the basis orthonormal. We describe necessary and sufficient conditions for the existence of a rank-one Einstein metric solvable extension of n. Since every two-step nilpotent Lie algebra attached to a graph (as introduced by S. G. Dani, M. G. Mainkar [Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc. 357 (2005) 2235--2251]) has such a nice basis, this note generalizes a recent result of H.-R. Fanaï [Einstein solvmanifolds and graphs, C. R. Acad. Sci. Paris, Ser. I 344 (2007) 37--39].