In the Book "Morse Theory" by J. Milnor in chapter 21 they study the structure of Riemannian Lie Groups with bi-invariant metrics.
For Left-Invariant vector fields $X,Y,Z,W$ and a bi-invariant riemannian metric $g$ it is quite straightforward to show that
Now he uses this relation to prove that the sectional curvature
$K(X,Y)=g([X,Y],[X,Y])=1/4g([X,Y],[X,Y]) \geq 0 $.
What i don't understand at this point, is why it is sufficient to look only at left-invariant vectorfields X,Y to prove that the sectional curvature is generally positive.
My guess is that is has to do with the bi-invariance of the metric and the fact that the space of left-invariant vector fields is isomorphic to the tangent Space at the identity. But i don't see how exactly i can make sense of that.