A graph is said to be {\em exceptional} if it is connected, has least eigenvalue greater than or equal to $-2$, and is not a generalized line graph. Such graphs are known to be representable in the root system $E_8$. The 473 maximal exceptional graphs were found initially by computer, and the 467 with maximal degree $28$ have subsequently been characterized. Here we use constructions in $E_8$ to prove directly that there are just six maximal exceptional graphs with maximal degree less than 28.