Editing Root system (section)

===The dual root system===
If Φ is a root system in ''E'', the '''coroot''' α<sup>∨</sup> of a root α is defined by
<math display=block>\alpha^\vee= {2\over (\alpha,\alpha)}\, \alpha.</math>

The set of coroots also forms a root system Φ<sup>∨</sup> in ''E'', called the '''dual root system''' (or sometimes ''inverse root system'').
By definition,  α<sup>∨ ∨</sup> = α, so that Φ is the dual root system of Φ<sup>∨</sup>. The lattice in ''E'' spanned by  Φ<sup>∨</sup> is called the ''coroot lattice''. Both Φ and Φ<sup>∨</sup> have the same Weyl group ''W'' and, for ''s'' in ''W'',
<math display=block> (s\alpha)^\vee= s(\alpha^\vee).</math>

If Δ is a set of simple roots for Φ, then Δ<sup>∨</sup> is a set of simple roots for Φ<sup>∨</sup>.<ref>{{harvnb|Hall|2015|loc=Proposition 8.18}}</ref>

In the classification described below, the root systems of type <math>A_n</math> and <math>D_n</math> along with the exceptional root systems <math>E_6,E_7,E_8,F_4,G_2</math> are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the <math>B_n</math> and <math>C_n</math> root systems are dual to one another, but not isomorphic (except when <math>n=2</math>).