Math. Z. 175, 139-141 (1980) Mathematische Zeitschrift 9 by Springer-Verlag 1980 A Geometric Property of Central Elements Maria Luiza Leite and Isabel Dotti de Miatello Departamento de Matemfitica, Universidade Federal de Pernambuco, 50000 Recife~Brazil Let G be a Lie group and g its associated Lie algebra. John Milnor shows in [1], Corollary 1.3, that if u belongs to the center of g and 1; is the sectional curvature associated with any left invariant metric on G then ~c(u,v)>0, for all v in g. He then asks whether central elements are the only ones satisfying that property. In this note we show that the answer to his question is affirmative. Notation (see [1], Sect. 5). Let G be an n-dimensional Lie group with Lie algebra g consisting of all smooth vector fields invariant under left translations (with the usual Lie bracket [, ] of vector fields). If (,) denotes a left invariant riemannian metric on G, we recall that the function (u, v) is constant for u, v e g; the riemannian connection V associated with ( , ) is given by ( v,,v, w)= l/2 { ([u, vl, w)-([v, w],u) + ([w,u],v) } ; the sectional curvature associated with an orthonormal pair {u,v} is the number and the Ricci curvature is given by r(u)= ~ 1r vi), where {v/}7= i i=1 is any orthonormal basis of g. Theorem. If x does not belong to the center of g, then there exists a left invariant metric and an element y s g so that to(x, y)< O. Proof If there is y6g such that the vectors x,y and Ix, y] are linearly inde- pendent then there exists a left invariant metric so that r(x)< 0 (see Theorem 2.5, [1]). Therefore •(x,z)<0 for some z in g. 0025-5874/80/0175/0139/$01.00