= Weyl integration formula
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Let $G$ be a <connected> <compact Lie group>, $T$ a <maximal torus>, and $W$ its <Weyl group>. With normalized <Haar measures> and $D(t)=\prod_{\alpha>0}(1-e^{-\alpha}(t))$, every <continuous> <class function> satisfies $\int_G f(g)\,dg=|W|^{-1}\int_T f(t)|D(t)|^2\,dt$. For a general <continuous function>, replace $f(t)$ by its average $\int_{G/T}f(gtg^{-1})\,d(gT)$. <Conjugation> from $G/T\times T$ covers the regular elements $|W|$ times, and its real <Jacobian determinant> is $|D(t)|^2$.
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