It's a classical fact that the commutative power series ring $\mathbb{Z}_p[[x_1,x_2]]$ is isomorphic to the completed group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times\mathbb{Z}_p]]$, the isomorphism sending generators $a_1,a_2$ of $\mathbb{Z}_p\times\mathbb{Z}_p$ to $1+x_1,1+x_2$.

In $\mathbb{Z}_p[[x_1,x_2]]$, we have the coset $1+(x_1,x_2)$ of the ideal $(x_1,x_2)$, which is a closed and open subgroup of its group of units, which certainly contain $1+x_1,1+x_2$, and hence contains the closed subgroup $\langle 1+x_1,1+x_2\rangle$ generated by $1+x_1,1+x_2$, which is isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_p$, but is strictly smaller than $1+(x_1,x_2)$.

Is it possible to describe the image of the closed subgroup $\langle 1+x_1,1+x_2\rangle$ in $\mathbb{Z}_p[[x_1,x_2]]$?

Ie, suppose we are given a power series $f\in 1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$. How can we determine if $f$ lies in the closed subgroup $\langle 1+x_1,1+x_2\rangle$?

(I know this question isn't really well-defined. I'm just looking for certain criteria, ideally linked to algebraic properties of the power series, which can be used to identify the image of $\langle 1+x_1,1+x_2\rangle$).

It would also be nice if one could describe the quotient $(1+( x_1,x_2))/\langle 1+x_1,1+x_2\rangle$.