  Bella\"iche has recently applied Pink-Lie theory to prove that, under mild
conditions, the image of a continuous 2-dimensional pseudorepresentation $\rho$
of a profinite group on a local pro-$p$ domain $A$ contains a nontrivial
congruence subgroup of ${\rm SL}_2(B)$ for a certain subring $B$ of $A$. We
enlarge Bella\"iche's ring and give this new $B$ a conceptual interpretation in
terms of conjugate self-twists of $\rho$, symmetries that naturally constrain
its image. As a corollary, this new $B$ is optimal among congruence subgroups
contained in the image. We also interpret the new $B$ vis-a-vis the adjoint
trace ring of $\rho$, which we show is a more natural ring for these questions
in general. Finally, we use our purely algebraic result to recover and extend a
variety of arithmetic big-image results for ${\rm GL}_2$ Galois representations
arising from elliptic, Hilbert, and Bianchi modular forms and $p$-adic Hida or
Coleman families of elliptic and Hilbert modular forms.
