Maybe it would clarify matters if I gave a little more background, in community wiki format.
The basic idea of this algebraic proof goes back to a short paper by Richard Brauer (1936) in German in Mathematische Zeitschrift, an interesting time in his life when he had been expelled from his professorship in Berlin and took up a position in Toronto. (This particular paper has no marginal additions, as do some other papers; I bought a used copy of his collected papers in three volumes which used to belong to him.)
Brauer's proof is of course less natural than the original proof by Weyl, but it applies to all semisimple Lie algebras over an algebraically closed field of characteristic 0 and may be the simplest algebraic proof. An attempt was made to simplify the proof, in a more recent textbook by K Erdmann and her recent student M. Wilden here published by Springer in 2006 as an undergraduate text. This result is not essential for their further work but occurs in Theorem 9.16 with an unconvincing proof. This was acknowledged by them in a list of errata, and affects Exercises 9.15 (cf. 9.16) as well perhaps as Lemma 10.7. (A good exercise is to track down the specific fault in the proof.) The book itself is perhaps an attractive alternative to mine, having a more leisurely pace and more examples but covering much less material in a similar number of pages.
The Brauer method is used in Bourbaki's Chapter 1 (of their treatise on Lie groups and Lie algebras) as well as Jacobson's 1962 book (now in a Dover reprint). But as stated above, Weyl's 1925 proof is more natural in the Lie group context, imitating the finite group method.