  In 1998, Liebeck and Seitz introduced a constant $t(G)$, dependent on the
root system of a reductive algebraic group $G$ and proved that if $x$ is a
semisimple element of order greater than $t(G)$ in $G$ then there exists an
infinite subgroup of $G$ stabilizing the same subspaces of $L(G)$ as $x$. The
values for $t(G)$ are $12$, $68$, $124$ and $388$ for $G=G_2,F_4,E_6,E_7$
respectively. In this paper we obtain a similar result for these groups and the
minimal module $V_{\mathrm{min}}$, obtaining significantly smaller numbers,
namely $4$, $18$, $27$ and $75$ respectively (with some small conditions on the
element $x$ that are not important for applications). Note that both $t(G)$ and
these new bounds are sharp. As a corollary we eliminate several potential
maximal subgroups $\mathrm{PSL}_2(q_0)$ of these groups that seem difficult to
eliminate through other means, along with other groups.
  This paper forms part of the author's programme to vastly reduce the number
of putative maximal subgroups of exceptional groups of Lie type.
