  Formulas describing all 2-element and 3-element factorizations of arbitrary
element of the groups SU(2) and SO(3,R) are derived. Six 2-element
factorizations, $ (U_{2}U_{3}U'_{2}), (U_{3}U_{2}U'_{3}), (U_{3}U_{1}U'_{3}),
(U_{1}U_{3}U'_{1}), (U_{1}U_{2}U'_{1}), (U_{2}U_{1}U'_{2})$, provide all
possible way to define Euler type angles; and six 3-element ones, $
(U_{1}U_{2}U_{3}), (U_{1}U_{3}U'_{2}), (U_{2}U_{3}U_{1}), (U_{2}U_{1}U_{3}),
(U_{3}U_{1}U_{2}), (U_{3}U_{2}U_{1})$ provide all possible ways to parameterize
the unitary and orthogonal groups by three elementary angles. In thecontext the
light polarization formalism of Stokes-Mueller vectors and Jones spinors,
relations produced give a base to resolve arbitrary pure polarization rotators
into all possible sets of elementary rotators of two or three constituents.
