= Ricci curvature of a bi-invariant Riemannian metric
{c}
{title2=$\operatorname{Ric}(X,X)=\tfrac14\sum_i\|[X,e_i]\|^2$}
For a <bi-invariant Riemannian metric> on a <Lie group>, the induced inner product on its <Lie algebra> makes every adjoint operator skew-adjoint. The <Levi-Civita connection of a bi-invariant metric> gives $R(X,Y)Z=-\tfrac14[[X,Y],Z]$. Thus $K(X,Y)=\tfrac14\|[X,Y]\|^2$ for orthonormal $X,Y$, and
$$
\operatorname{Ric}(X,X)=\frac14\sum_i\|[X,e_i]\|^2.
$$
The nullspace of this quadratic form is exactly the <center of a Lie algebra>. A zero center therefore gives positive <Ricci curvature>, uniformly bounded below by a positive constant times the metric through left invariance and compactness of the unit sphere in the <Lie algebra>.
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