PBW bases and centralisers for the $q$-Onsager algebra

We prove that, over an arbitrary field and whenever $q$ is not a root of unity, the Baseilhac–Kolb root vectors form a PBW basis of the $q$-Onsager algebra for every total order on the positive roots of ${\widehat{\mathfrak{s}\mathfrak{l}}}_2$. This removes the previous transcendence hypothesis. We establish twelve PBW bases in the alternating generators and show that they persist under arbitrary scalar central specialisation of the alternating central extension. We determine the centraliser of the negative alternating subalgebra and that of the first imaginary alternating generator, and deduce that the four single-family alternating polynomial subalgebras are maximal commutative. Together, these results settle four conjectures of Terwilliger and, in characteristic different from $2$, a conjecture of Baseilhac and Belliard.