If $G$ is a maximal exceptional graph then either (a) $G$ is the cone over a graph switching-equivalent to the line graph $L(K_8)$ or (b) $G$ has $K_8$ as a star complement for the eigenvalue $-2$ (or both). In case (b) it is shown how $G$ can be constructed from $K_8$ using intersecting families of $3$-sets.