  We propose a geometric explanation for the observation that generic quadratic
polynomials over split quaternions may have up to six different factorizations
while generic polynomials over Hamiltonian quaternions only have two. Split
quaternion polynomials of degree two are related to the coupler motion of
"four-bar linkages" with equal opposite sides in universal hyperbolic geometry.
A factorization corresponds to a leg of the four-bar linkage and during the
motion the legs intersect in points of a conic whose focal points are the fixed
revolute joints. The number of factorizations is related by the number of real
focal points which can, indeed, be six in universal hyperbolic geometry.
