Schur's Lemma of the Version for Lie Algebras

Question

In general, Schur's lemma is stated as follows.

(i) Let $(\pi_1,V_1)$ and $(\pi_2,V_2)$ be irreducible representations, and let $T:V_1\rightarrow V_2$ be an intertwining operator. Then either $T$ is zero or it is an isomorphism.

(ii) Suppose that $(\pi,V)$ is an irreducible representation of $G$ and $T:V\rightarrow V$ is an intertwining operator. Then there exists a scalar $\lambda\in\mathbb{C}$ such that $T(v)=\lambda v$ for all $v\in V$.

Now I'm considering the Lie algebras' version of Schur's lemma in Exercise 10.5 in Lie Groups by Daniel Bump.

Let $\mathfrak{g}$ be a Lie algebra, and let $V$, $W$ be simple $\mathfrak{g}$-modules.

(i') The space of $\mathfrak{g}$-module homomorphisms $\phi:V\rightarrow W$ is at most one-dimensional.

(ii') The space of invariant bilinear forms $V\times W\rightarrow\mathbb{C}$ is at most one-dimensional.

Here are my idea and question to verify Schur's lemma of the version for Lie Algebras:

One can conclude (i') immediately from (i) if there exists an irreducible representation for any Lie algebra (how to demonstrate it or where can I find the result about it?). We may obtain (ii') by a proposition in Bump's Lie Groups:

Proposition 10.5. Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over a field $F$. Then there exists, up to scalar, at most one invariant bilinear form on $\mathfrak{g}$. If a nonzere invariant bilinear form exists it is nondegenerate and symmetric.

Can we generalize Proposition 10.5 in the case that $\mathfrak{g}$ is an arbitrary Lie algebra to conclude (ii')? Moreover, what's the relation between the above two versions of Schur's lemma?

Answer 1

Thanks for the comments. We may add some information to the original exercise, that $V$ and $W$ should be simple $\mathfrak{g}$-modules.

For (i'), since simple $\mathfrak{g}$-module is already an irreducible representation, by (i), the space of $\mathfrak{g}$-module homomorphisms is at most one-dimensional.

For (ii'), since a bilinear form $V\times W\rightarrow \mathbb{C}$ gives a $\mathfrak{g}$-linear map from $V$ to the dual of $W$, and it follows that it is an intertwining operator. Then applying (i) we can conclude what we desire.