In the case of SU(3) (or its complexified Lie algebra, formula_35), we may choose a base consisting of two roots formula_36 at an angle of 120 degrees, so that the third positive root is formula_37. In this case, the element formula_25 is the reflection about the line perpendicular to formula_39. Then the map formula_32 is the reflection about the line "through" formula_39. The self-dual representations are then the ones that lie along the line through formula_39. These are the representations with labels of the form formula_43, which are the representations whose weight diagrams are "regular" hexagons.