  A description of the embedding of the universal Askey--Wilson algebra, AW(3),
in $U_q(sl_2)^{\otimes 3}$ is given in terms of the universal R-matrix of
$U_q(sl_2)$. The generators of the centralizer of $U_q(sl_2)$ in its three-fold
product are naturally expressed through conjugations of Casimir elements with
R. They are identified as the images of the generators of AW(3) under the
embedding map by showing that they obey the AW(3) relations. This is achieved
by introducing a natural coaction also constructed with the help of the
R-matrix.
