In do-carmo's Book "Riemannian Geometry" there is an exercise on proving existence of a bi-invariant metric on any compact connected Lie group. (pg 46, question 7).
In the first stage, you are required to prove that every differential form (of top degree) which is left invariant, is also right invariant.
However, I do not see where we are using this property later. The basic method of the proof is this: take any left invariant metric on $G$ , and use integration of right translates of it to produce a bi-invariant metric.
The proof of the bi-invariance of the constructed metric does not use the fact $\omega$ is bi-invariant, but only its left-invariance. (More specifically, the left invariance of the metric is implied by the left invariance of the original metric taken, and the right invariance follows by using left invariance of $\omega$ together with the fact that the integral of a form does not change under pullback via orientation preserving diffeomorphism).
So, I am a bit puzzled by the first stage. Is it really needed for the proof?