How to proof that the $\mathbb{Z}$-span of weights of a faithful $L$-modul contains the root lattic?

Question

Let $L$ be a semisimple Lie Algebra with root system $\Phi$ and base $\Delta$ of $\Phi$. Let $V$ be a finit dimensional, faithful $L$-modul with weights $\Pi(V)$.
I am trying to show that the $\mathbb{Z}$-span of $\Pi{(V)}$ (denoted $\Lambda(V)$) lies between $\Lambda$, the lattic of integral linear functions, and $\Lambda_{r}$, the root lattic.
The inclusion $\Lambda(V) \subseteq \Lambda$ is obvious, since all weights are integral linear functions and $\mu(h_{i})=\left<\mu,\alpha_{i} \right>$, $\alpha_{i}\in\Delta$ is linear in the first variable.
I am struggeling with the second inclusion $\Lambda_{r} \subseteq \Lambda(V)$.
I tried the following:
First of all let $\alpha\in\Phi$ be arbitrary. It suffices to show that either $\alpha$ or $-\alpha$ lie in $\Lambda(V)$, because then does the whole $\mathbb{Z}$-span of them. To do so I chose $\mu\in\Pi(V)$ and took a look at the $\alpha$-string throught $\mu$.
If $\mu+\alpha$ is a weights it is $\mu+\alpha-\mu=\alpha\in\Lambda(V)$ since $-\mu\in\Lambda(V)$.
Otherwise I did the same with $\mu-\alpha$.
This would show, that all $\stackrel{+}{-}\alpha$ lie in $\Lambda(V)$ and so does $\Lambda_{r}$, but I am not sure if this is anywhere near correct and I have no clue where the fact that $V$ is faithful should come in to play.
Thank you for helping me.

Answer 1

$V$ being a faithful ${\mathfrak g}$-module means that the structure homomorphism $\rho: {\mathfrak g}\to{\mathfrak g}{\mathfrak l}(V)$ is injective. By definition, $\rho$ is a homomorphism of Lie algebras, but we may also view it as a morphism of ${\mathfrak g}$-modules if we equip ${\mathfrak g}$ with the adjoint action and ${\mathfrak g}{\mathfrak l}(V)$ with the ${\mathfrak g}$-action given by $X.- := [\rho(X),-]$.

Now your claim follows from the following observations: Firstly, by the injectivity of ${\mathfrak g}\to {\mathfrak g}{\mathfrak l}(V)$ any root of ${\mathfrak g}$ (i.e. any weight of the adjoint representation of ${\mathfrak g}$) is a weight of ${\mathfrak g}{\mathfrak l}(V)$. Secondly, ${\mathfrak g}{\mathfrak l}(V)\cong V\otimes_{\mathbb k} V^{\ast}$ as ${\mathfrak g}$-modules, and weights add upon tensoring. Can you fill in the details yourself?