Pauli-coordinate Haar measure on SU(2) ($d\rho =d^{3}u/|u_0|$)

The hemisphere graph $u_0=\pm \sqrt{1-|\mathbf{u}{|}^2}$ of SU(2) as the three-sphere has induced metric $g_{ij}={\delta }_{ij}+u_i{u}_j/u_0^2$, whose determinant is $1/u_0^2$. Its volume density is therefore $d^{3}u/|u_0|$. Fixed left or right multiplication preserves the Euclidean norm on the four Pauli coefficients, so the round volume is Haar measure. Both hemispheres have volume ${\pi }^2$, giving total volume $2{\pi }^2$ and normalized density $d^{3}u/(2{\pi }^2|u_0|)$ on each hemisphere. The apparent equatorial divergence is an integrable coordinate singularity.