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Motivated by this question we ask:

Is there any relation between the Ricci curvature of a Lie group and the killing form of its Lie algebra?Under what conditions, they are proportional to each other?

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When the Lie group is compact and semisimple, then the Killing form $B$ is negatively definite and $-B$ defines a bi-invariant metric. Its Ricci curvature is $$Ric(X)=-\frac{1}{4}\sum_{i=2}^nB(\left[X,e_i\right],\left[X,e_i\right])$$ for any $B$-orthonormal base $\left\{e_1=X,e_2,\ldots,e_n\right\}$ with $e_1=X$. In particular, $Ric(X)$ is nonnegative.

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