  A novel generalization of the Askey-Wilson algebra is presented and shown to
be associated with coproducts in the quantum algebra $U_q(su(1,1))$. This
algebra has 15 non-commuting generators given by $Q^{(A)}$, with $A\subset
\{1,2,3,4\}$ and their 5 linearly independent inversions generated by the
algebra automorphism $q\rightarrow q^{-1}$, $E\leftrightarrow F.$ The set of
generators can be split into operators fixed under inversion, and those with an
orientation under this inversion. We then show that the generators will either
commute or satisfy q-commutator relations linear in the generators, with the
restriction that parity operators commute with generators of opposite parity
and the q-commutation relations are between those with the same parity.
Finally, we give a novel algebra expression satisfied by the generators
involving only the natural generators, i.e. those arising from the coupling
scheme.
