Serre-Lusztig relations for $\imath$quantum groups III

Abstract: let $\widetilde{\bf U}^\imath$ be a quasi-split universal $\imath$quantum group associated to a quantum symmetric pair $(\widetilde{\bf U}, \widetilde{\bf U}^\imath)$ of Kac-Moody type with a diagram involution $\tau$. We establish the Serre-Lusztig relations for $\widetilde{\bf U}^\imath$ associated to a simple root $i$ such that $i \neq \tau i$, complementary to the Serre-Lusztig relations associated to $i=\tau i$ which we obtained earlier. A conjecture on braid group symmetries on $\widetilde{\bf U}^\imath$ associated to $i$ disjoint from $\tau i$ is formulated.