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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 $G$ suchthat $T$ 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, weshow 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 theelements $w$ that have a lift in $N(G,T)$ of order $|w|$.

On splitting of the normalizers of maximal tori in $E_7(q)$ and $E_8(q)$