Exercise 10, § 4

a) There exists an irreducible representation $\pi$ of orthogonal type of the alternating group $\mathfrak{A}_5$ such that $M_4(\pi ) = 3$.

b) Let R be a root system of type $\mathbf{E}_6$ (resp. of type $\mathbf{E}_7$, of type $\mathbf{E}_8$) and W the Weyl group of R. Let $\pi$ be the geometric representation of W (LIE, V, § 4, No.$^o3$). It is irreducible, of orthogonal type, and satisfies $M_4(\pi ) = 3$. (One may use software such as GAP or Magma to determine the list of conjugacy classes of W and their images under $\pi .$)