On the images of certain $G_2$-valued automorphic Galois representations

In this paper we study the images of certain families ${\rho_{\pi,\ell} }_\ell$of$G_2$-valuedGaloisrepresentationsof$\mbox{Gal}(\overline{F}/F)$associatedto$L$-algebraic regular, self-dual, cuspidal automorphic representations $\pi$ of $G{L}_7({\mathbb{A}}_F)$, where $F$ is a totally real field. In particular, we prove that, under certain automorphic conditions, the images of the residual representations ${\bar{\rho }}_{\pi ,\mathrm{\ell }}$ are as large as possible for infinitely many primes $\mathrm{\ell }$. Moreover, we apply our result to some examples constructed by Chenevier, Renard and Ta"ibi.