Exercise 12, § 4

Let G be a connected semisimple compact Lie group whose complexified Lie algebra is isomorphic to the simple Lie algebra $\mathfrak{e}_6$.

a) There exists a faithful irreducible representation $\varrho$ of G of dimension 27; any such representation is of complex type.

b) The fourth moment of the representation $\varrho$ is equal to 3.