Orientability of Lie Group

Question

I am trying to show that a Lie Group $G$ is orientable. I would like to do this by construccting a nowhere vanishing top form $\omega$ on $G$, which would then imply that $G$ is orientable.

I think that the general idea should be to somehow find a nowhere vanishing form at a point (probably the identity?) and then somehow "shift" the form around $G$ by left multiplication, but I am not sure about how the details would work. Could someone help me this more precise?

Answer 1

Every Lie group is parallelizable, then orientable. Just consider $\psi:G \times T_1G \rightarrow TG$ given by $\psi(g,v)=(g, d(L_g)_1(v))$.