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In this paper we prove the following result.  Let $G$ be a simply connected simple linear algebraic group of exceptionalLie type over an algebraically closed field $F$ of characteristic $p\geq 0$,and let $u\in G$ be a nonidentity unipotent element. Let $\phi$ be anon-trivial irreducible representation of $G$.  Then the Jordan normal form of $\phi(u)$ contains at most one non-trivialblock if and only if $G$ is of type $G_2$, $u$ is a regular unipotent elementand $\dim \phi\leq 7$.  Note that the irreducible representations of the simple classical algebraicgroups in which a non-trivial unipotent element is represented by a matrixwhose Jordan form has a single non-trivial block were determined by I.D.Suprunenko (Unipotent elements of non-prime order in representations of theclassical algebraic groups: two big Jordan blocks, J. Math. Sci. 199(2014), 350-- 374.

Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block