  The Killing form \beta\ of a real (or complex) semisimple Lie group G is a
left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein
metric. Let \Omega\ denote the multiple of its curvature operator, acting on
symmetric 2-tensors, with the factor chosen so that \Omega\beta=2\beta. The
result of Meyberg [8], describing the spectrum of \Omega\ in complex simple Lie
groups G, easily implies that 1 is not an eigenvalue of \Omega\ in any real or
complex simple Lie group G except those locally isomorphic to SU(p,q), or
SL(n,R), or SL(n,C) or, for even n only, SL(n/2,H), where p\ge q\ge0 and
p+q=n>2. Due to the last conclusion, on simple Lie groups G other the ones just
listed, nonzero multiples of the Killing form \beta\ are isolated among
left-invariant Einstein metrics. Meyberg's theorem also allows us to understand
the kernel of \Lambda, which is another natural operator. This in turn leads to
a proof of a known, yet unpublished, fact: namely, that a semisimple real or
complex Lie algebra with no simple ideals of dimension 3 is essentially
determined by its Cartan three-form.
