Conformal Properties of Indefinite Bi-Invariant Metrics

Conformally Einstein Lorentzian Lie groups: extensions of the Euclidean and Poincaré groups

“…Remark 1.10. Conformally Einstein and Bach-flat bi-invariant indefinite metrics have been considered in [19], where it is shown that any such metric on a solvable Lie group is Bach-flat with two-step nilpotent Ricci operator. Moreover, since an inner product on a Lie algebra (g, •, • ) extends to a bi-invariant metric on the corresponding Lie group if and only if [x, y], z = x, [y, z] , one has that none of the metrics in theorems 1.1, 1.5, and 1.8 is biinvariant.…”