  This paper is devoted to analize inside the infinitely many possible bases of
Uq(g), same that can be considered "more equal then others". The element of
selection has been a privileged relation with the bialgebra. A new parameter z'
has been found that determines the commutation relations, independent from the
z=log(q) that defines Uq(g). Both z and z' are necessary to fix the relations
between the basic set and its coproducts. Three cases are particularly
relevant: the analytical set with z'=z, the Lie set with Lie-like commutation
relations (for z'=0) and the canonical/crystal basis with z' infinity. To
simplify the exposition, we discuss in details the easy generalizable case of
Uq(su(2)).
