  We complete the classification of maximal representations of uniform complex
hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional
groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $\rho$ is a maximal
representation of a uniform complex hyperbolic lattice $\Gamma\subset{\rm
SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm
E}_6$, and we describe completely the representation $\rho$. The case of
classical Hermitian target groups was treated by Vincent Koziarz and the second
named author (arxiv:1506.07274). However we do not focus immediately on the
exceptional cases and instead we provide a more unified perspective, as
independent as possible of the classification of the simple Hermitian Lie
groups. This relies on the study of the cominuscule representation of the
complexification of the target group. As a by-product of our methods, when the
target Hermitian group $G$ has tube type, we obtain an inequality on the Toledo
invariant of the representation $\rho:\Gamma\rightarrow G$ which is stronger
than the Milnor-Wood inequality (thereby excluding maximal representations in
such groups).
