Kempe factorizations for rational curves on ${\mathrm{SO}}_4(\mathbb{R})$

We study constructive Kempe factorizations for rational curves on ${\mathrm{SO}}_4(\mathbb{R})$. Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree $2d$ with $d\ge 1$ first factors into $d$ quadratic rational curves and then into a product of at most $2d$ planar rotation curves, where each planar rotation curve fixes a two-dimensional plane pointwise. The construction proceeds by extracting left and right isoclinic polynomial parts via Cayley’s factorization, factoring the corresponding quaternion polynomials, and pairing linear factors with equal norm polynomials. The resulting algorithms are explicit and are illustrated by examples.