If a g-module is sum of irreps then is direct sum of irreps

Question

In Infinite dimensional Lie algebras book by Victor G Kac, In prop.3.6 He proves that, any integrable $g(A)$ - module $V$ is direct sum of finite dimensional, irreducible, $h$ - invariant $g_{(i)}$ modules. He has proved only that $V$ is sum of such irreducibles, but he hasnt given the proof for the sum is direct . How to prove the sum is indeed direct.

Is it a general fact in module theory?

Any help is welcomed.

Thanks in Advance

Answer 1

Yes, this is a standard fact. Zorn's Lemma implies there is a submodule $W$ of $V$, such that $W$ is a direct sum of irreducibles, but no larger submodule of $V$ is a direct sum. The maximality implies that $W = V$. Otherwise, there is some irreducible $L$ that is not contained in $W$, so $L + W$ contradicts the maximality of $W$ (because the irreducibility implies $L \cap W = \{0\}$, so $L + W = L \oplus W$).