Completely reducible Lie module implies existence of a complementar module

Question

The following is the exercise 6.2 in Humphreys' Introduction to Lie Algebra and Representation Theory. I wish to know if my attempt of the 'only if' direction is correct.

Suppose $V$ is a direct sum of irreducible sub-modules $V_1,\dots,V_n$. Take a submodule $W$ of $V$, we have that $W \cap V_i$ is a submodule of $V_i$ hence $V_i$ is $0$ or $V_i$. Take $I=\{i: V_i \cap W=0 \}$ and define $W'$ as

$$W'=\bigoplus_{i \in I} V_i,$$

this subspace does the job: we have direct sum $V=W \oplus W'$ and that $W'$ is a submodule of $V$.

I'm asking this question because every other argument I have found on the web seems a bit dissimilar, so I think I'm missing something.

Answer 1

I realized why the proof is incorrect. We can’t say $V=W \oplus W’$, the intersection $W \cap W’$ may be non zero a priori:

$$W \cap W’=W \cap (\oplus_{i \in I} V_i) \neq \oplus_{i \in I}(W \cap V_i)=0$$