Motivated by [this question](https://mathoverflow.net/questions/31198/curvature-of-a-lie-group) 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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Ricci curvature and killing form
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.


