*''X'' <supmath>*X^\ast</supmath> and ''X''<submath>*X_\ast</submath> are free abelian groups of finite [[Rank (linear algebra)|rank]] together with a [[perfect pairing]] between them with values in '''<math>\mathbb{Z'''}</math> which we denote by ( , ) (in other words, each is identified with the [[dual lattice]] of the other).
*& <math>\Phi;</math> is a finite subset of ''X''<supmath>*X^\ast</supmath> and Φ<supmath>v\Phi^\vee</supmath> is a finite subset of ''X''<submath>*X_\ast</submath> and there is a bijection from &<math>\Phi;</math> onto Φ<supmath>v\Phi^\vee</supmath>, denoted by α→α<supmath>v\alpha\mapsto\alpha^\vee</supmath>.
* For each &<math>\alpha;</math>, <math>(&\alpha;, &\alpha;<sup>v</sup>^\vee)=2</math>.
* For each &<math>\alpha;</math>, the map taking ''<math>x''\mapsto to ''x''−-(''x'',&\alpha^\vee)\alpha;<sup>v</supmath>)α induces an automorphism of the root datum (in other words it maps &<math>\Phi;</math> to &<math>\Phi;</math> and the induced action on ''X''<submath>*X_\ast</submath> maps Φ <supmath>v\Phi^\vee</supmath> to Φ<supmath>v\Phi^\vee</supmath>)
The elements of &<math>\Phi;</math> are called the '''roots''' of the root datum, and the elements of Φ<supmath>v\Phi^\vee</supmath> are called the '''coroots'''. The elements of <math>X^\ast</math> are sometimes '''[[Weight_(representation_theory)|weights]]''' and those of <math>X_\ast</math> accordingly '''coweights'''.
If &<math>\Phi;</math> does not contain <math>2&\alpha;</math> for any &<math>\alpha; \in &\Phi;</math>, then the root datum is called '''reduced'''.
