Monodromy and irreducibility of type A₁ automorphic Galois representations

Let K be a totally real field and \ρπ,λ:GalKₙ( E_λ)_λ the strictly compatible system of K defined over E attached to a regular algebraic polarized cuspidal automorphic representation π of GLₙ( AK). Let G_λ be the algebraic monodromy group of ρπ,λ. If there exists λ₀ such that (a) ρπ,λ₀ is irreducible, (b) Gλ₀ is connected and of type A₁, and (c) either at most one basic factor in the exterior tensor decomposition of the tautological representation of Gλ₀ᵈᵉʳ is (SL₂,std) or the tautological representation of Gλ₀ᵈᵉʳ is (SO₄,std), we prove that Gλ, Cn, C is independent of λ, ρπ,λ is irreducible for all λ and residually irreducible for almost all λ. If moreover K= Q, we prove that the compatible system \ρπ,λ_λ is up to twist constructed from some two-dimensional modular compatible systems.