Rational motions in conformal three space are naturally represented by curves on the spin group $\operatorname{Spin}(4,1)$, which projects to $\operatorname{SO}_{4,1}$. This connects the study of rational motions in conformal geometric algebra to the decomposition of curves on real classical groups. We prove that a rational curve on $\operatorname{SO}_4(\mathbb{R})$ of degree greater than $2$ can be decomposed into a product of rational curves of degree $2$ only in the shape of planar rotations by a constructive decomposition algorithm. The first step in the algorithm begins by applying the Cayley's factorization of 4D rotations, which allows the rational curve on $\operatorname{SO}_4(\mathbb{R})$ to be expressed as a product of two isoclinic curves on $\operatorname{SO}_4(\mathbb{R})$. By a homomorphism from the group of isoclinic curves to the group of motion polynomials, which have quaternion coefficients, the second step is to completed by replacing the decomposition problem for isoclinic curves by decomposing a motion polynomial into linear factors. The product of two adjacent linear factors corresponds to a rational curve of degree $2$, thereby giving the desired decomposition as a product of planar rotations and constant matrices. With the decomposition of rational curves on $\operatorname{SO}_4(\mathbb{R})$ established, the corresponding result for $\operatorname{SE}_4(\mathbb{R})$ follows directly.