= Killing-form Einstein metric
{c}
{title2=$g=B,\quad\operatorname{Ric}=-\tfrac14B$}
= Killing metric
{c}
{synonym}
On a real semisimple <Lie group>, the nondegenerate invariant <Killing form> gives a <bi-invariant pseudo-Riemannian metric>. For left-invariant fields the <Koszul formula> gives $\nabla_XY=[X,Y]/2$ and $R(X,Y)Z=-[[X,Y],Z]/4$. Taking the trace gives $\operatorname{Ric}(Y,Z)=-B(Y,Z)/4$, proving the Einstein property. On a compact <semisimple group>, $-B$ is positive definite and has Einstein constant $1/4$. The indefinite general construction is not a proof of existence of a positive-definite bi-invariant metric on a noncompact group.
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