  We consider bivariate polynomials over the skew field of quaternions, where
the indeterminates commute with all coefficients and with each other. We
analyze existence of univariate factorizations, that is, factorizations with
univariate linear factors. A necessary condition for existence of univariate
factorizations is factorization of the norm polynomial into a product of
univariate polynomials. This condition is, however, not sufficient. Our central
result states that univariate factorizations exist after multiplication with a
suitable univariate real polynomial as long as the necessary factorization
condition is fulfilled. We present an algorithm for computing this real
polynomial and a corresponding univariate factorization. If a univariate
factorization of the original polynomial exists, a suitable input of the
algorithm produces a constant multiplication factor, thus giving an a
posteriori condition for existence of univariate factorizations. Some
factorizations obtained in this way are of interest in mechanism science. We
present an example of a curious closed-loop mechanism with eight revolute
joints.
