To be specific, I have four 4-spheres, that is, for each of the spheres, we have a relation of the form, \begin{equation} x_1^2 +x_2^2 +x_3^2 +x_4^2 =1. \end{equation} So, I have twelve degrees of freedom in total. Further, I have a function defined over these four 4-spheres. I can, of course, parameterize each of the spheres individually using hyperspherical coordinates. But are there coordinate systems developed that would allow me to treat the twelve variables in some "global/unified" fashion? Of course, the product of two 2-spheres (circles) is a torus. So, it would seem that I am looking for some generalization/extension of coordinates tailored to a torus.
The twelve-variate function in which I am interested in studying, arose in the course of computing the (Moore) determinant of a $4 \times 4$ Hermitian matrix with its off-diagonal entries being quaternions, except for the (1,2)-(2,1) and (3,4)-(4-3) pairs which are set to zero.