Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar curvature R. 



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> **Question** [Edited] Is there a nice formula which expresses the scalar curvature at a point of the manifold in terms of the lie algebra of the group?
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Curvature of a Lie group
See Exercice 1 in Chapter 4 of Do Carmo's "Riemannian Geometry". 


The formula is $R(X,Y)Z = \frac 1 4 [[X,Y], Z]$.


In particular, if $X$ and $Y$ are orthonormal, the sectional curvature of the generated plane is 


$K(\sigma)= \frac 1 4 \|[X,Y]\|^2$ 


Which is always $\geq 0$. 


EDIT: In view of the comments, it is important to add that this is for a bi-invariant metric. 


