On irreducibility of certain low dimensional automorphic Galois representations

We study irreducibility of Galois representations $ρ_{π,λ}$ associated to a $n=7$ or 8-dimensional regular algebraic essentially self-dual cuspidal automorphic representation $π$ of $\text{GL}_n(\mathbb{A}_\mathbb{Q})$. We show $ρ_{π,λ}$ is irreducible for all but finitely many $λ$ under the following extra conditions. (i) If $n=7$, and there exists no $λ$ such that the Lie type of $ρ_{π,λ}$ is the standard representation of exceptional group $\textbf{G}_2$. (ii) If $n=8$, and when there exist infinitely many $λ$ such that the Lie type of $ρ_{π,λ}$ is the spin representation of $\text{SO}_7$, we assume there exist no three distinct Hodge-Tate weights form a 3-term arithmetic progression.