By definition, if $\Phi$ and $\Phi'$ are root systems of the Euclidean spaces $E$ and $E'$, respectively, then an isomorphism $\Phi\to\Phi'$ is one that is induced by an isomorphism $E\to E'$ which preserves the inner product on pairs of roots, but is not necessarily an isometry.
I read that since the axioms of a root system are unchanged by scaling the inner products be a positive real number, we can assume that the isomorphism is induced by an isometry.
How is this possible? It seems like this suggests that given an isomorphism $\psi\colon E\to E'$, it's possible to scale the inner products on $E$ and $E'$ so that $s(\psi(x),\psi(y))=r(x,y)$ for all $x,y\in E$ and some fixed $s,r>0$. But this doesn't seem like it'd be true at all.