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