  Both uniaxial and biaxial nematic liquid crystals are defined by
orientational ordering of their building blocks. While uniaxial nematics only
orient the long molecular axis, biaxial order implies local order along three
axes. As the natural degree of biaxiality and the associated frame, that can be
extracted from the tensorial description of the nematic order, vanishes in the
uniaxial phase, we extend the nematic director to a full biaxial frame by
making use of a singular value decomposition of the gradient of the director
field instead. New defects and degrees of freedom are unveiled and the
similarities and differences between the uniaxial and biaxial phase are
analyzed by applying the algebraic rules of the quaternion group to the
uniaxial phase.
