Before revision:  We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases.
After revision:  We begin with the 4D data problem for at least five decades in several different bodies of scientific literature where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; we focus on these exact solutions to expose the structure of the entire eigensystem for the traditional 3D and 2D data problems as special cases.