 Longest Weyl-group element

The longest element of a finite Weyl group is the element sending all positive roots of a root system to negative roots. It is the longest element of a finite Coxeter group. For the G2 root system it is $-I$: its twelve roots form two hexagons, and the dihedral reflection group of a root system contains the rotation through $\pi$. It determines the highest weight of a dual representation.