= Bi-invariant pseudo-Riemannian metric
On a connected <Lie group>, a nondegenerate invariant symmetric <bilinear form> on a <Lie algebra> extends by translation to a metric preserved by both left and right translation. For a disconnected <Lie group>, the form must additionally be invariant under the full <Adjoint representation of a Lie group>. It may have indefinite signature. Its <Levi-Civita connection> is $\nabla_XY=\frac12[X,Y]$ for <left-invariant vector fields>, and its <Ricci tensor> is $-K/4$. Choosing the nondegenerate <Killing form> of a semisimple <Lie algebra> yields a canonical indefinite <Einstein manifold>; compact groups instead admit the positive metric $-K$.
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