= Root system
{wiki}
In the general reflection convention, a root system in a finite-dimensional real <inner product space> $E$ is a finite spanning set $\Phi\subset E\setminus\{0\}$ such that every <root reflection>
$$
s_\alpha(v)=v-\frac{2(v,\alpha)}{(\alpha,\alpha)}\alpha
$$
permutes $\Phi$. In particular $-\alpha\in\Phi$ whenever $\alpha\in\Phi$. A <reduced root system> additionally requires $\Phi\cap\mathbb R\alpha=\{\alpha,-\alpha\}$. A <crystallographic root system> additionally requires all <Cartan integers> $2(\beta,\alpha)/(\alpha,\alpha)$ to be <integers>. Some authors include these extra conditions in the term root system; they are stated separately here.
For example, the <A2 root system> is $\{e_i-e_j:i\ne j\}$ in the plane $x_1+x_2+x_3=0$. Its six <roots> form a regular hexagon, and their <reflections> generate a <Weyl group> isomorphic to $S_3$. The <roots> of a complex <semisimple Lie algebra> form a reduced crystallographic root system: its <root-space decomposition> converts infinitesimal <Lie brackets> into this finite Euclidean geometry. The <Dynkin diagram> then records enough simple-root data to classify its simple components.<Lie Groups and Lie Algebras by Pavel Etingof (2024)>[Sections 19–23, root spaces, root systems, and Dynkin diagrams]{ref}
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