I was wondering whether the triviality of the tangent bundle of a Lie group is a shared feature with all other vector bundles over Lie groups.
Naturally, the first example of a non-trivial vector bundle that comes to mind is the Moebius strip, which is a vector bundle over $S^1$ -a Lie group-, so the answer is no. However, the tangent bundle of a Lie group is also a group, and is hence orientable (unlike the Moebius strip).
So, my question is: Are orientable vector bundles over Lie groups trivial? or actually better, does anybody know an example of a non-trivial vector bundle over a Lie group which is itself a Lie group?
I thought that a way around this question was to inspect the Euler class of the given bundle, because I thought that I knew the cohomology of Lie groups. I had Borel's theorem in mind:
"If $G$ is a compact connected Lie group, then the cohomology ring is an exterior algebra of odd degree"
I thought that this implied that even rank bundles would be necessarily trivial. Unfortunately, later I realized this doesn't say that there are no elements of even degree... Also, this would only work for compact groups, and I would like to say something of non-compact groups as well.