Positive systems and simple roots

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system (Reduced crystallographic Euclidean root system). A vector $v\in E$ is regular (for $\mathrm{\Phi }$) if $(v,\alpha )\mathrm{\ne }0$ for every $\alpha \in \mathrm{\Phi }$; such vectors exist because $\mathrm{\Phi }$ is finite, the finitely many hyperplanes ${\alpha }^{\perp }$ are proper subspaces of the finite-dimensional real vector space $E$, and $E$ is not the union of finitely many proper subspaces (A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces).

Fix a regular $v\in E$. A root $\alpha \in \mathrm{\Phi }$ is positive (with respect to $v$) if $(v,\alpha )>0$, and negative if $(v,\alpha )<0$. Write ${\mathrm{\Phi }}^{+}=\{\alpha \in \mathrm{\Phi }:(v,\alpha )>0\},{\mathrm{\Phi }}^{-}=\{\alpha \in \mathrm{\Phi }:(v,\alpha )<0\},$ so that $\mathrm{\Phi }={\mathrm{\Phi }}^{+}\bigsqcup {\mathrm{\Phi }}^{-}$ and ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$; every root is positive or negative, since $v$ is regular. A positive root $\alpha \in {\mathrm{\Phi }}^{+}$ is simple if it is not a sum $\alpha =\beta +\gamma$ of two positive roots $\beta ,\gamma \in {\mathrm{\Phi }}^{+}$; we write $\mathrm{\Delta }$ for the set of simple roots.

The set $\mathrm{\Delta }$ depends on the choice of the regular vector $v$; the subsequent theorem proves that $\mathrm{\Delta }$ is a basis of $E$ and that every root is an integral combination of $\mathrm{\Delta }$ whose nonzero coefficients all have one sign.