Einstein nilradicals are completable

Abstract

An Einstein nilradical is a nilpotent Lie algebra which can be the nilradical of an Einstein metric solvable Lie algebra. A Lie algebra is called complete if its center is zero and all its derivations are inner. A nilpotent Lie algebra is called completable if it is the maximal nilpotent ideal of a complete solvable Lie algebra. In this short note, based on the classification result of Einstein metric solvable Lie algebras, we show that any Einstein nilradical is completable. This provides a purely algebraic obstruction for a nilpotent Lie algebra to be an Einstein nilradical.