  In this note we prove that the explicit realization of arbitrary complex
powers of generators of quantum group $U_{q}(\mathfrak{sl}(2))$ satisfies all
the commutation relations of the algebra of complex powers, including the
generalized Kac's identity which was announced in our previous paper. It turns
out that the latter identity in this realization is equivalent to $6-9$
integral identity on quantum dilogarithm.
