  Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the
$q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements
recursively, and it is obscure how to express them in closed form. To mitigate
the difficulty, we bring in the universal Askey-Wilson algebra $\Delta_q$.
There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to
\Delta_q$. We apply $\natural $ to the above PBW basis, and express the images
in closed form. Our results make heavy use of the Chebyshev polynomials of the
second kind.
