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 understood the following steps:

- 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 $W/W'$.
- We proceed by induction on the 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 $\tilde{W}$.
- We suppose that $W$ is irreducible, so we may use Schur's lemma on $c \vert_{W}$ to say that $\operatorname{Ker} c$ is an $L$-submodule of $V$, where $c$ is the endomorphism of $V$ defined in 6.2.

The other parts of the proof are very hard, and 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?

wherein the book this is?), but it seems odd without context to start with talk of codimension 1 submodules, as there is a priori no reason why any codimension 1 subspace should be a sub module. $\endgroup$special casewhere $V$ has an $L$-submodule $W$ of codimension 1 which allows use of a previous lemma. $\endgroup$1more comment