Exercise 14, § 4

Let G be a connected compact semisimple Lie group. Suppose that there exists a faithful irreducible representation $\varrho$ of dimension 27 of G. The complexified Lie algebra of G is isomorphic to $\mathfrak{e}_6$ if and only if there exists an irreducible subrepresentation of dimension 78 in the representation End($\varrho$ ).