Let $R$ and $R'$ be irreducible root systems in the real inner product spaces $E$ and $E'$. Prove that $R$ and $R'$ are isomorphic iff there exists a scalar $\lambda \in \mathbb{R}$ and a vector space isomorphism $\varphi: $ $E \to E'$ such that $\varphi(R)=R'$ and $(\varphi(u),\varphi(v))=\lambda(u,v)\text{ for all }u,v \in E$. (Introduction to Lie algebra Erdmann Karin- Mark Wildon, Exercise 11.15, page 124.)
I just can prove the "if" part and I get stuck with the "only if part". I highly appreciate who can give me some ideas.
Thank in advance
Two root systems $R$ and $R’$ are said to be isomorphic if there is a vector space isomorphism $\phi : E \to E’$ such that:
(1) $\phi(R) = R’$
(2) for any two roots $\alpha$ and $\beta \in R$, $\langle \alpha, \beta \rangle = \langle \phi(\alpha), \phi(\beta) \rangle$.
Now, for the only if direction, you only need to prove that $\lambda(u,v) = (\phi(u), \phi(v))$. Notice that if $u,v \in R$ this follows easily from condition (2), by definition of $\langle u,v \rangle$ and $\langle \phi(u), \phi(v) \rangle$.