A module of a Lie algebra can be decomposed into irreducibles if and only if every submodule has a complement.

Question

Let $V$ be an $L$-module of a lie algebra $L$ (A representation of $L$). Prove that $V$ can be written as the direct sum of irreducible submodules if and only if every $L$-submodule $W \subseteq V$ has a complement.

I have an attempt at the solution but I am new to this subject so I have no confidence that it is correct. This is an exercise from Humphreys textbook on Lie algebra.

The forward direction: Let $W$ be an $L$-submodule of $V$. Then we have the exact sequence of vector spaces.

$$0 \to W \to V \to V/W \to 0$$

Note that $V/W$ is the set complement of $W$. This is an exact sequence of $L$-Modules if $V/W$ is a sub representation of $V$. By hypothesis this is true. By the splitting lemma, this short exact sequence of $L$-modules splits if the map from $V$ to $V/W$ has a section. Choose this section to be the inclusion of $V/W$ into $V$. Therefore this sequence splits and $V \cong W \oplus V/W$. Repeat this process until $V$ and $V/W$ have no submodules. I am not sure why this process should terminate.

The backward direction is relatively trivial. If $V \cong W_1 \oplus \ldots \oplus W_n$ where the $W_i$ for $i \in \{1,\ldots n\}$ are irreducible then every submodule corresponds to some choice irreducibles in the above direct product. The complement of those submodules correspond to the direct product of every irreducible you didn't choose.

Answer 1

The most important thing is the first convention in the beginning of the chapter

In this section all representations are finite dimensional, unless otherwise noted. (Humphreys, Chp 6)

Thus your main concern about whether it terminates vanishes. I do not think it is true if $V$ is not finite dimensional, but I do not have a counter example.

Answer 2

Your proof for the direction "every submodules of $V$ has a complement $\Rightarrow$ $V$ decomposes ..." is wrong on surprisingly many levels. It is neither true that $V/W$ is a set complement of $W$, nor that $V/W$ is a subrepresentation of $V$ (it is isomorphic to one though), nor that that would be equivalent to the sequence being an exact sequence of $L$-modules (that is always true with the right definition of $L$-action on the quotient), nor that that is true by hypothesis (rather, turns out the hypothesis is equivalent to every such sequence being split).

Rather, replace all these wrong claims by:

Let $W$ be an $L$-submodule of $V$. By assumption in this direction, it has a complement $W'$, meaning by definition that there is an $L$-submodule $W'$ of $V$ such that $V \simeq W \oplus W'$.

From here, continue with the other answer and its comments.