Serre-Lusztig relations for $imath$quantum groups III

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 $i eq tau i$, 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.