# Weyl's theorem and Representations

Let $$L$$ be a semisimple Lie algebra and let $$(V,\varphi)$$ be a finite-dimensional $$L−$$module representation. Our main goal is to prove that $$\varphi$$ is completely reducible. Consider an $$L−$$submodule of $$V$$ of codimension one, let $$0 \longrightarrow W \longrightarrow V \longrightarrow F \longrightarrow 0$$ be an exact sequence (Where $$F$$ is an $$L−$$ module). From the book of James Humphreys called "Introduction to Lie Algebras and Representation Theory", i have understand the following steps:

1. We take another proper submodule of W denoted by W′ such that the exact sequence $$0 \longrightarrow W/W′ \longrightarrow V/W′\longrightarrow F \longrightarrow 0$$ splits, so there exists a one dimensional $$L−$$submodule of $$V/W′$$ (say $$\tilde{W}/W′$$) complementary to $$W/W′$$.

2.We proceed by induction on dimension of $$W$$, so we get an exact sequence $$0 \longrightarrow W′ \longrightarrow \tilde{W} \longrightarrow F \longrightarrow 0$$ which splits. It follows easily that $$V=W \oplus X$$ where X is a submodule complementary to $$W′$$ in $$\tilde{W}$$.

3.We suppose that $$W$$ is irreducible, so we may use Schur's lemma on $$c \vert_{W}$$ to say that $$Ker \; c$$ is an $$L−$$ submodule of $$V$$, where $$c$$ is an endomorphism of $$V$$ defined in 6.2.

The other parts of the proof are very hard, i didn't understand them. Can someone help me to figure out those parts? If there is another comprehensible method, can someone share it with us?