Right-invariance of a volume form on a compact Lie group

differential-geometry lie-groups riemannian-geometry

The claim is simply false in general. I think, Lee forgot to assume that the group is connected or that the orientation is bi-invariant. As a simple example, as you suggested, consider $G=O(2)$. Then the action of $G$ on itself via conjugation does not preserve any orientation on $G_0=SO(2)$ (since this action contains a reflection). From this, it follows that there is no bi-invariant orientation. From this, it follows that there is no bi-invariant volume form. Of course, there is a bi-invariant measure on $G$, given by a bi-invariant density on $G$.

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