*For each <math>\alpha</math>, we have: <math> \langle \alpha, \alpha^{\vee}\rangle =2 </math>
*For each &<math>\alpha;</math>, the map taking<math>x \mapsto ''x''to- \langle ''x''−(''x'',&\alpha;<sup>v^{\vee} \rangle \alpha </supmath>)α induces an automorphism of the root datum (in other words it maps Δ<math>\Psi</math> to Δ<math>\Psi</math> and the induced action on ''X''<submath>*X^{\vee} </submath> maps Δ <supmath>v \Psi^{\vee} </supmath> to Δ<sup>v</sup>)itself.
The elements of <math>\Psi</math> are called the '''roots''' of the root datum, and the elements of <math> \Psi^{\vee} </math> are called the '''coroots'''.
