Exercise 17, § 6

Show that the algebra derived from a real simple Lie algebra by extending the base field from $\mathbf{R}$ to $\mathbf{C}$ is not always simple. (Let $g$ be a real simple Lie algebra. If $g' = g_{(\mathbf{C})}$ is simple, let $h$ be the real Lie algebra derived from $g'$ by restricting the field of scalars to $\mathbf{R}$. We know that $h$ is simple. Then $h_{(\mathbf{C})}$ is, by Exercise 4 of § 1, the product of two algebras isomorphic to $g'$. Cf. also Exercise 26 (b).)