For \(\ell\)-adic Galois representations \(\rho_\ell\) on the étale cohomology of smooth proper varieties over a number field \(K\) it is conjectured that the image is a reductive \(\ell\)-adic Lie group, in some sense “independent of \(\ell\)”. J. P. Serre has at least shown that the group of connected components of the algebraic hull is independent of \(\ell\), defining a finite Galois extension \(K^{\text{conn}}\) of \(K\). It is contained in all the fixed fields \(K_\ell\) of \(\ker(\rho_\ell)\). The main result of the paper states that \(K^{\text{conn}}= \bigcap_\ell K_\ell\), where \(\ell\) runs over a set of primes which has relative Dirichlet density one in the set of all rational primes splitting in a fixed number field \(E\). The proof uses Serre’s techniques as well as some Chebotarev arguments.