Mapping a finite 2D space onto a 3D sphere is relatively trivial:
$$X_3 \in [-1, 1), \quad\phi \in [0, 2 \pi) \\ X_1 = \cos(2 \pi \phi) \sqrt{1 - X_3^2} \\ X_2 = \sin(2 \pi \phi) \sqrt{1 - X_3^2}$$
Schnabel and Janke (2022) call it the polar method, but it is referenced in other places too – even called too obvious to require a reference. I could, however, not find anywhere how this generalizes to higher dimensions. Specifically, how such a mapping would look for the 4-sphere, i.e., (uniformly) mapping 4 parameters to the surface of a 5D unit hypersphere.
Every source I could find only points to the fact, that, for an n-sphere, a vector with (n+1) normal distributed elements, normalized for lengths one, is uniformly distributed. This is true, yet this requires (n+1) parameters to cover what is an n-dimensional surface – one parameter more than ought to be necessary.
Schnabel, S., & Janke, W. (2022). A simple algorithm for uniform sampling on the surface of a hypersphere (arXiv)