A new approach to the $\mathrm{ı}$Serre and Serre-Lusztig relations for $\mathrm{ı}$quantum groups

We give a new, conceptual proof of the $\mathrm{ı}$Serre and Serre-Lusztig relations for $\mathrm{ı}$quantum groups. The key to our approach is a new formula for the comultiplication of the $\mathrm{ı}$-divided powers, which allows us to reformulate the relations in terms of the adjoint action. We then obtain a proof using properties of the adjoint representation. The flexibility of this approach allows us to establish a more general family of relations which seem difficult to establish otherwise.