I am trying to solve the following exercise from Humphreys:
Let $\Phi$ be a root system, let $\Lambda_r$ and $\Lambda$ be its root lattice and weight lattice respectively. Prove that any subgroup $\Lambda^\prime$ of $\Lambda$ including $\Lambda_r$ is $\mathscr{W}$-invariant, where $\mathscr{W}$ is the Weyl group of $\Phi$. Therefore, we obtain a homomorphism $\phi:\mathrm{Aut}\Phi/\mathscr{W}\to\mathrm{Aut}(\Lambda/\Lambda_r)$. Prove $\phi$ is injective, then deduce that $-1\in\mathscr{W}$ if and only if $\Lambda\supset 2\Lambda$.
Question: I have proved everything except the assertion that $\phi$ is injective.
Some idea: Since $\mathrm{Aut}\Phi/\mathscr{W}\simeq\{\sigma\in\mathrm{Aut}\Phi:\sigma(\Delta)=\Delta\}$, where $\Delta$ is a base of $\Phi$ fixed previously, then $\sigma$ in $\mathrm{Aut}\Phi/\mathscr{W}$ is a permutation of the set $\Delta$, and $\sigma$ permutes correspondently on the set of fundamental dominant weights relative to $\Delta$. If $\sigma\in\mathrm{Aut}\Phi/\mathscr{W}$ is nontrivial, then it sends some fundamental dominant weight $\lambda_i$ to the another, say $\lambda_j$. I want to prove $\lambda_i-\lambda_j\notin\Lambda_r$, then $\phi(\sigma)$ is nontrivial in $\mathrm{Aut}(\Lambda/\Lambda_r)$.
The claim is right? How we can prove it, or just prove directly that $\phi$ is injective?
I feel sorry for my poor English. Any help would be appreciated!