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Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism$L$ of the $q$-Onsager algebra $\mathcal O_q$. In this paper, we express eachof $L, L^{-1}$ as a formal sum involving some quantum adjoints. In addition,(i) we give a computer-free proof that $L$ exists; (ii) we establish the higherorder $q$-Dolan/Grady relations previously conjectured by Baseilhac and ThaoVu; (iii) we obtain a Lusztig automorphism for the current algebra $\mathcalA_q$ associated with $\mathcal O_q$; (iv) we describe what happens when afinite-dimensional irreducible $\mathcal O_q$-module is twisted via $L$.

The Lusztig automorphism of the $q$-Onsager algebra