Monodromy of four dimensional irreducible compatible systems of Q

Let $F$ be a totally real field and $n\le 4$ a natural number. We study the monodromy groups of any $n$-dimensional strictly compatible system $\{{\rho }_{\lambda }{\}}_{\lambda }$ of $\lambda$-adic representations of $F$ with distinct Hodge-Tate numbers such that ${\rho }_{{\lambda }_0}$ is irreducible for some ${\lambda }_0$. When $F=\mathbb{Q}$, $n=4$, and ${\rho }_{{\lambda }_0}$ is fully symplectic, the following assertions are obtained. (i) The representation ${\rho }_{\lambda }$ is fully symplectic for almost all $\lambda$. (ii) If in addition the similitude character ${\mu }_{{\lambda }_0}$ of ${\rho }_{{\lambda }_0}$ is odd, then the system $\{{\rho }_{\lambda }{\}}_{\lambda }$ is potentially automorphic and the residual image ${\bar{\rho }}_{\lambda }({\text{Gal}}_{\mathbb{Q}})$ has a subgroup conjugate to ${\text{Sp}}_4({\mathbb{F}}_{\mathrm{\ell }})$ for almost all $\lambda$.