  We obtain a stochastic differential equation (SDE) satisfied by the first $n$
coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$.
The SDE has non-Lipschitz coefficients but we are able to provide an analysis
of existence and pathwise uniqueness and show that they always hold. The square
of the radial component is a Wright-Fisher diffusion with mutation and it
features in a skew-product decomposition of the projected spherical Brownian
motion. A more general SDE on the unit ball in $\mathbb{R}^{n+\ell}$ allows us
to geometrically realize the Wright-Fisher diffusion with general non-negative
parameters as the radial component of its solution.
