  In this paper we prove that for any connected reductive algebraic group G and
a large enough prime $l$, there are continuous homomorphisms
$$\mathrm{Gal}(\bar\mathbb Q/\mathbb Q) \to G(\bar\mathbb Q_l)$$ with
Zariski-dense image, in particular we produce the first such examples for
$SL_n, Sp_{2n}, Spin_n, E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a
mod-$l$ representation of $\mathrm{Gal}(\bar\mathbb Q/\mathbb Q)$ related to
the Weyl group of $G$ and use a variation of Stefan Patrikis' generalization of
a method of Ravi Ramakrishna to deform it to characteristic zero.
