An attractive basis for the $q-$Onsager algebra

Let $\mathsf{\text{A}},{\mathsf{\text{A}}}^{\ast }$ be the fundamental generators of the $q-$Onsager algebra. A linear basis for the $q-$Onsager algebra is known as the `zig-zag’ basis [IT09]. In this letter, an attractive basis for the $q-$Onsager algebra is conjectured, based on the relation between the $q-$Onsager algebra and a quotient of the infinite dimensional algebra ${\mathcal{A}}_q$ introduced in [BK05].