  Let $F$ be a totally real field and $n\leq 4$ 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 $\text{Sp}_4(\mathbb{F}_\ell)$ for almost all $\lambda$.
