  In this paper we prove the following result.
  Let $G$ be a simply connected simple linear algebraic group of exceptional
Lie 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 a
non-trivial irreducible representation of $G$.
  Then the Jordan normal form of $\phi(u)$ contains at most one non-trivial
block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element
and $\dim \phi\leq 7$.
  Note that the irreducible representations of the simple classical algebraic
groups in which a non-trivial unipotent element is represented by a matrix
whose Jordan form has a single non-trivial block were determined by I.D.
Suprunenko (Unipotent elements of non-prime order in representations of the
classical algebraic groups: two big Jordan blocks, J. Math. Sci. 199(2014), 350
-- 374.
