= Crystallographic root system
{title2=$2(\beta,\alpha)/(\alpha,\alpha)\in\mathbb Z$}
A finite <root system> in a real <inner-product space> is crystallographic when every <Cartan integer> $2(\beta,\alpha)/(\alpha,\alpha)$ is an integer. The roots of a complex <semisimple Lie algebra> satisfy this because each number is a weight of its <sl2 subalgebra associated with a root>. Reducedness is a separate axiom; the <root-space reducedness lemma> establishes it for these Lie-algebra roots.
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