Rank and isomorphism of root systems

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system (Reduced crystallographic Euclidean root system).

Its rank is $\mathrm{rank}\mathrm{\Phi }={\dim }_{\mathbb{R}}E$. Since $\mathrm{\Phi }$ spans $E$, the rank is determined by $\mathrm{\Phi }$; it is the number of simple roots of any base of $\mathrm{\Phi }$.

Let ${\mathrm{\Phi }}^{\mathrm{\prime }}\subseteq E^{\mathrm{\prime }}$ be a second reduced crystallographic root system. An isomorphism of root systems $\phi :\mathrm{\Phi }\to {\mathrm{\Phi }}^{\mathrm{\prime }}$ is a linear isomorphism $\phi :E\to E^{\mathrm{\prime }}$ with $\phi (\mathrm{\Phi })={\mathrm{\Phi }}^{\mathrm{\prime }}$ that preserves every Cartan integer: for all $\alpha ,\beta \in \mathrm{\Phi }$, $\frac{2(\phi (\beta ),\phi (\alpha ))}{(\phi (\alpha ),\phi (\alpha ))}=\frac{2(\beta ,\alpha )}{(\alpha ,\alpha )}\mathrm{.}$

Because $\frac{2(\beta ,\alpha )}{(\alpha ,\alpha )}=2\cos \theta \frac{\mathrm{\mid }\beta \mathrm{\mid }}{\mathrm{\mid }\alpha \mathrm{\mid }},\theta \text{ the angle between }\alpha ,\beta ,$ a map as above preserves the angle of every pair of nonproportional roots and the ratio of their lengths whenever the two roots lie in the same irreducible component; conversely, a linear isomorphism carrying $\mathrm{\Phi }$ onto ${\mathrm{\Phi }}^{\mathrm{\prime }}$ that preserves the angle and the length ratio of every pair of nonproportional roots preserves all Cartan integers and is therefore an isomorphism of root systems. An isomorphism need not preserve the given inner products on the nose, but within each irreducible component it preserves the common scale, hence all angles between roots and all ratios $\mathrm{\mid }\beta \mathrm{\mid }\mathrm{/}\mathrm{\mid }\alpha \mathrm{\mid }$ formed by two roots of the same irreducible component; ratios of lengths of roots taken from different irreducible components need not be preserved.