A new PBW basis for the alternating central extension of the $q$-Onsager algebra

We establish a new PBW basis for ${\mathcal{A}}_q$, the alternating central extension of the $q$-Onsager algebra. Terwilliger showed that the alternating generators form a PBW basis in the block order $\mathcal{G}<{\mathcal{W}}^{-}<{\mathcal{W}}^{+}<\tilde{\mathcal{G}}$. We prove that they also form a PBW basis in the different block order ${\mathcal{W}}^{-}<\mathcal{G}<\tilde{\mathcal{G}}<{\mathcal{W}}^{+}$. Consequently, multiplication induces a vector-space isomorphism $${\mathcal{W}}^{-}\otimes \mathcal{G}\otimes \tilde{\mathcal{G}}\otimes {\mathcal{W}}^{+}⟶{\mathcal{A}}_q,$$ thereby confirming a conjecture of Terwilliger.