Proposition 1

Let $R_1$ and $R_2$ be two reduced root systems in vector spaces $V_1$ and $V_2$. Let $B_1 = (\alpha_i)_{i \in I_1}$ and $B_2 = (\alpha_i)_{i \in I_2}$ be bases of $R_1$ and $R_2$, indexed canonically. Let $\lambda$ be an isomorphism from the Dynkin graph of $R_1$ to the Dynkin graph of $R_2$. Then, there exists a unique isomorphism from $V_1$ to $V_2$ transforming $R_1$ into $R_2$ and $\alpha_i$ into $\alpha_{\lambda(i)}$ for all $i \in I_1$.

It is clear that an automorphism of $R$ defines an automorphism of the Dynkin graph of $R$, and hence a homomorphism $\varphi$ from the group $A(R)$ to the group of automorphisms of the Dynkin graph of $R$.