I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant find any useful literature how this works. Especially, I am confused since the Lie algebra seems to always be defined around the unity, but I guess on a group manifold one can use that we can always map any point to unity and thus also map the tangent space?
So Let's say I have a Lie group manifold parametrized by coordinates x0,x1,...,xn. How can I obtain the metric induced from the killing form?