Monodromy and irreducibility of type A₁ automorphic Galois representations

Let K be a totally real field and \,: GalKₙ (E_) \_ the strictly compatible system of K defined over E attached to a regular algebraic polarized cuspidal automorphic representation of GLₙ (AK). Let G_ be the algebraic monodromy group of,. If there exists ₀ such that (a), 䃐 is irreducible, (b) G䃐 is connected and of type A₁, and (c) either at most one basic factor in the exterior tensor decomposition of the tautological representation of G䃐^der is (SL₂, std) or the tautological representation of G䃐^der is (SO₄, std), we prove that G, ₂₍, ₂ is independent of, , is irreducible for all and residually irreducible for almost all. If moreover K= Q, we prove that the compatible system \, \_ is up to twist constructed from some two-dimensional modular compatible systems.