Conjugation Jacobian for a compact Lie group ($J(t)={\prod }_{\alpha >0}|1-\alpha (t){|}^2$)

For a connected compact Lie group with maximal torus $T$, the map $q:G/T\times T\to G$, $q(xT,t)=xt{x}^{-1}$, has the displayed real Jacobian determinant. Choose an invariant inner product and identify the tangent space of $G/T$ with ${\mathfrak{t}}^{\perp }$. Translation of the derivative at $(T,t)$ to the identity gives $(X,Y)\mapsto (\mathrm{Ad}(t^{-1})-I)X+Y$. Each pair of opposite roots of a root system corresponds to a real two-plane on which the determinant is $|1-\alpha (t){|}^2$. The torus direction has determinant one. Normalized quotient and Haar measures have matching volume constants because the Riemannian submersion gives $\mathrm{vol}G=\mathrm{vol}(G/T)\mathrm{vol}T$. The map covers regular elements $|W|$ times, giving the Weyl integration formula with conjugacy averaging for general functions.