applies, where is the set of positive roots, is the highest weight, and is half the sum of the positive roots. The inner product formula_150 is that of the Lie algebra formula_127 invariant under the action of the Weyl group on formula_152 the Cartan subalgebra. The roots (really elements of formula_153 are via this inner product identified with elements of formula_154 For formula_138 the formula reduces to , "where the present notation must be taken into account". The highest weight is . By taking tensor products, the result follows.