  In this paper, we discuss maximality of Seidel matrices with a fixed largest
eigenvalue. We present a classification of maximal Seidel matrices of largest
eigenvalue $3$, which gives a classification of maximal equiangular lines in a
Euclidean space with angle $\arccos1/3$. Motivated by the maximality of the
exceptional root system $E_8$, we define strong maximality of a Seidel matrix,
and show that every Seidel matrix achieving the absolute bound is strongly
maximal.
