Exercise 13, § 4

Let $G$ be a connected semisimple compact Lie group and $\mathfrak{g}$ its Lie algebra. Suppose that there exists a faithful irreducible representation $\varrho$ of dimension 27 of $G$. Suppose that the fourth moment of $\varrho$ is equal to 3 and that the representation $\varrho$ is of complex type.

a) The Lie algebra $\mathfrak{g}$ is simple.

b) The Lie algebra $\mathfrak{g}$ is isomorphic to $\mathfrak{e}_6$.