  Let $K$ be a totally real field and
$\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline
E_\lambda)\}_\lambda$ the strictly compatible system of $K$ defined over $E$
attached to a regular algebraic polarized cuspidal automorphic representation
$\pi$ of $\mathrm{GL}_n(\mathbb A_K)$. Let $\mathbf G_\lambda$ be the algebraic
monodromy group of $\rho_{\pi,\lambda}$. If there exists $\lambda_0$ such that
(a) $\rho_{\pi,\lambda_0}$ is irreducible, (b) $\mathbf G_{\lambda_0}$ is
connected and of type $A_1$, and (c) either at most one basic factor in the
exterior tensor decomposition of the tautological representation of $\mathbf
G_{\lambda_0}^{\mathrm{der}}$ is $(\mathrm{SL}_2,\mathrm{std})$ or the
tautological representation of $\mathbf G_{\lambda_0}^{\mathrm{der}}$ is
$(\mathrm{SO}_4,\mathrm{std})$, we prove that $\mathbf G_{\lambda,\mathbb
C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $\lambda$,
$\rho_{\pi,\lambda}$ is irreducible for all $\lambda$ and residually
irreducible for almost all $\lambda$. If moreover $K=\mathbb Q$, we prove that
the compatible system $\{\rho_{\pi,\lambda}\}_\lambda$ is up to twist
constructed from some two-dimensional modular compatible systems.
