  Let $G$ be a finite group of Lie type $E_l$ with $l\in\{6,7,8\}$ over $F_q$
and $W$ be the Weyl group of $G$. We describe all maximal tori $T$ of $G$ such
that $T$ has a complement in its algebraic normalizer $N(G,T)$. Let $T$
correspond to an element $w$ of $W$. When $T$ does not have a complement, we
show that $w$ has a lift in $N(G,T)$ of order $|w|$ in all considered groups,
except the simply-connected group $E_7(q)$. In the latter case we describe the
elements $w$ that have a lift in $N(G,T)$ of order $|w|$.
