Formulae of $\mathrm{ı}$-divided powers in ${\mathbf{U}}_q({\mathfrak{s}\mathfrak{l}}_2)$, II

The coideal subalgebra of the quantum ${\mathfrak{s}\mathfrak{l}}_2$ is a polynomial algebra in a generator $t$ which depends on a parameter $\kappa$. The existence of the $\mathrm{ı}$-canonical basis (also known as the $\mathrm{ı}$-divided powers) for the coideal subalgebra of the quantum ${\mathfrak{s}\mathfrak{l}}_2$ were established by Bao and Wang. We establish closed formulae for the $\mathrm{ı}$-divided powers as polynomials in $t$ and also in terms of Chevalley generators of the quantum ${\mathfrak{s}\mathfrak{l}}_2$ when the parameter $\kappa$ is an arbitrary $q$-integer. The formulae were known earlier when $\kappa =0,1$.