  We construct Zariski-dense surface subgroups in infinitely many
commensurability classes of uniform lattices of the split real Lie groups
$\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$,
$\operatorname{SO}(k+1,k)$, and $\operatorname{G}_2$. These subgroups are
images of Hitchin representations. In particular, we show that every uniform
lattice of $\operatorname{Sp}(2n,\mathbb{R})$, of $\operatorname{SO}(k+1,k)$
with $k\equiv1,2[4]$ and of $\operatorname{G}_2$ contains infinitely many
mapping class group orbits of Zariski-dense Hitchin representations of fixed
genus. Together with Long-Thistlethwaite and with a previous paper of the
author, it implies that all lattices of $\operatorname{Sp}(4,\mathbb{R})$
contain a Zariski-dense surface subgroup.
