Very simple 2-adic representations and hyperelliptic Jacobians

Let $K$ be a field of characteristic zero, $n\ge 5$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $\mathrm{S}_n$ or the alternating group $\mathrm{A}_n$. Let $C\colon y^2 = f(x)$ be the corresponding hyperelliptic curve and $X = J(C)$ its Jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(X)$ for the $\mathbf{Q}_{\ell}$-Tate module of $X$ and $e_{\lambda}$ for the Riemann form on $V_{\ell}(X)$ attached to the theta divisor. Let $\mathfrak{sp}(V_{\ell}(X),e_{\lambda})$ be the $\mathbf{Q}_{\ell}$-Lie algebra of the symplectic group of $e_{\lambda}$. Let $\mathfrak{g}_{\ell,X}$ be the $\mathbf{Q}_{\ell}$-Lie algebra of the image of the Galois group $\mathrm{Gal}(K)$ of $K$ in $\mathrm{Aut}(V_{\ell}(X))$. Assuming that $K$ is finitely generated over $\mathbb{Q}$, we prove that $\mathfrak{g}_{\ell,X}=\mathbf{Q}_{\ell}\operatorname{Id}\oplus \mathfrak{sp}(V_{\ell}(X),e_{\lambda})$ where $\operatorname{Id}$ is the identity operator.