Weyl's theorem

《Introduction to Lie Algebras and Representation Theory》 · 第 15、28 页 · Chapter II 的底域约定及 Section 6.3「Weyl's Theorem」的 Theorem（省略中间无关文字）

Virtually all of the remaining theory to be developed in this book will require the assumption that $F$ have characteristic $0$. (Some of the exercises will indicate how counterexamples arise in prime characteristic.) Moreover, in order to have available the eigenvalues of $\operatorname{ad}x$ for arbitrary $x$ (not just for $\operatorname{ad}x$ nilpotent), we shall assume that $F$ is algebraically closed, except where otherwise specified. […] Theorem (Weyl). Let $\phi:L\to\mathfrak{gl}(V)$ be a (finite dimensional) representation of a semisimple Lie algebra. Then $\phi$ is completely reducible.