= Canonical torsion-free connection on a Lie group
{title2=$\nabla^0_XY=\frac12[X,Y]$}
The average of the two <flat translation connections on a Lie group> is a <torsion-free connection>. For <left-invariant vector fields>, the <Jacobi identity> gives $R(X,Y)Z=-\frac14[[X,Y],Z]$ with $R(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$. Contracting gives $\operatorname{Ric}(X,Y)=-\frac14K(X,Y)$ for the <Killing form> $K$. It is flat exactly when all double Lie brackets vanish, including the case of a two-step nilpotent group. For a semisimple <Lie algebra>, $K$ is nondegenerate and supplies a bi-invariant indefinite metric whose <Levi-Civita connection> is $\nabla^0$.
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