Corollary

Let $R_1$ and $R_2$ be two reduced root systems in vector spaces $V_1$ and $V_2$, and let $B_1$ and $B_2$ be bases of $R_1$ and $R_2$. Let $f : B_1 \to B_2$ be a bijection that transforms the Cartan matrix of $R_1$ to that of $R_2$. Then there exists an isomorphism $F : V_1 \to V_2$ that transforms $R_1$ to $R_2$ and $\alpha$ to $f(\alpha)$ for all $\alpha \in B_1$.

Let $F$ be the isomorphism from $V_1$ to $V_2$ that takes $\alpha$ to $f(\alpha)$ for all $\alpha \in B_1$. Then $F$ transforms $s_\alpha$ to $s_{f(\alpha)}$, hence $W(R_1)$ to $W(R_2)$ (Th. 2), and hence $R_1$ to $R_2$ (Prop. 15).