I saw the following exercise on Knapp's famous textbook 'Representation Theory of Semisimple Groups. It's exercise 3 of chapter 1.
Suppose $\mathfrak{ g}$ is a complex simple Lie algebra ( of finite dimension). First regard it as a real Lie algebra, then consider its complexification. The exercise is: prove that after these operations, the obtained complex Lie algebra is not simple.
I tried to write down a set of basis and consider their explicit changes to find a nontrivial ideal. ( Because I had no idea which theorem I could use directly.) But I didn't succeed.
Any hint or suggestion would be welcome. Thanks a lot in advance!
