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Let $G$ be a compact Lie group and $\left\langle ,\right\rangle $ be a left invariant metric on $G$; $\omega$ be a positive differential $n$-form on $G$ which is left invariant. Consider the metric $(,)$ on $G$ given by: $$ \left(u,v\right)=\int_{G}\left\langle (dR_{x})_{y}u,(dR_{x})_{y}v\right\rangle _{yx}\omega$$ It's not too hard to show that this is left-invariant but I'm wondering how to show that $\left(,\right)$ is right-invariant?

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