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Formulas describing all 2-element and 3-element factorizations of arbitraryelement of the groups SU(2) and SO(3,R) are derived. Six 2-elementfactorizations, $ (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 allpossible 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 parameterizethe unitary and orthogonal groups by three elementary angles. In thecontext thelight polarization formalism of Stokes-Mueller vectors and Jones spinors,relations produced give a base to resolve arbitrary pure polarization rotatorsinto all possible sets of elementary rotators of two or three constituents.

Factorizations for 3-rotations and polarization of the light in Mueller-Stokes an Jones formalisms