Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation ρ of a profinite group on a local prop domain A contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We enlarge Bellaïche's ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of ρ, symmetries that constrain its image, and in terms of the adjoint trace ring of ρ, which we show is both more natural and the optimal 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 GL2 Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms.