Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If is an element of the Weyl group of a semisimple Lie algebra, then . In the case of formula_138 the Weyl group is . It follows that each is isomorphic to its dual formula_166 The root system of formula_167 is shown in the figure to the right. The Weyl group is generated by formula_168 where formula_169 is reflection in the plane orthogonal to as ranges over all roots. Inspection shows that so . Using the fact that if are Lie algebra representations and , then , the conclusion for is