Let $L$ be a finite dimensional semisimple Lie algebra with Cartan subalgebra $H$, $\Phi$ the roots of $L$, $\Lambda_r = \mathbb{Z}\Phi$ the root lattice, $\Lambda_w$ the weight lattice and $\Lambda(V) = \mathbb{Z}\{\text{weights of $V$}\}$ the subgroup of $\Lambda_w$ generated by all the weights.
Let $V$ be a Lie-module with corresponding representation $\rho :L \rightarrow \mathfrak{gl}(V)$, and suppose it is faithful. I want to show that $\Lambda_r \subseteq \Lambda(V)$.
I equipped $L$ with the adjoint representation and $\mathfrak{gl}(V)$ with the $L$-action $X.- = [\rho(X),-]$ thus making $\rho$ a Lie-module homomorphism.
Let $\alpha \in \Phi$, $h\in H$, then $L_\alpha \neq 0$ so there exists some $x\in L$ such that $[hx]=\alpha(h)x = h.x$, therefore roots of $L$ are weights of $L$ with the adjoint representation.
Furthermore, $\rho(x)\neq 0$ due to $\rho$ being faithful, so $$h.\rho(x) = [\rho(h),\rho(x)] = \rho([hx]) = \rho(\alpha(h)x) = \alpha(h)\rho(x)$$ So $\alpha$ is also a weight of $\mathfrak{gl}(V)$ with the $L$-action defined above.
Now it remains to show that $\alpha$ is a $\mathbb{Z}$-linear combination of weights of $V$. We know that $\mathfrak{gl}(V) \cong V^*\otimes V$, so let $(f\otimes v) \in V^* \otimes V$, $w\in V$, $h\in H$ and suppose $\lambda$ is the weight of $(f\otimes v)$
After some compution (which might be wrong, but I have a broken thumb so I won't type it all) I got
$$(h.(f\otimes v))(w) = f(w)\rho(h)v-f(\rho(h)w)v = \lambda(h)f(w)(v)$$
and I can't see how to proceed, or even if this is a good approach. Any help would be appreciated.
Also, I would like to show that if $\Lambda'$ is another subgroup of $\Lambda_w$ containing $\Lambda_r$, then $\Lambda' = \Lambda(V)$. Could you drop a hint on showing that?
Thank you in advance!