Root system

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 reflectionpermutes $\Phi$. 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.^[ref]