Weyl integration formula for SU(2) (${\int }_{\mathrm{SU}(2)}F(g)dg=\frac{2}{\pi }{\int }_0^{\pi }F(\mathrm{diag}(e^{i\theta },e^{-i\theta })){\sin }^2\theta d\theta$)

For a class function on SU(2) and normalized Haar measure, the displayed formula reduces a group integral to its conjugacy-class angle. The weight follows by identifying SU(2) with the unit sphere in four real dimensions: writing a unit quaternion as $\cos \theta +\mathbf{n}\sin \theta$ gives spherical volume proportional to ${\sin }^2\theta d\theta d\mathbf{n}$. Since ${\int }_0^{\pi }{\sin }^2\theta d\theta =\pi /2$, normalization gives $2/\pi$.