If, for a given Lie algebra, formula_17 is "not" in the Weyl group, then the dual of an irreducible representation will generically not be isomorphic to the original representation. To understand how this works, we note that there is always a unique Weyl group element formula_25 mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber. Then if we have an irreducible representation with highest weight formula_26, the "lowest" weight of the dual representation will be formula_27. It then follows that the "highest" weight of the dual representation will be formula_28. Since we are assuming formula_17 is not in the Weyl group, formula_25 cannot be formula_17, which means that the map formula_32 is not the identity. Of course, it may still happen that for certain special choices of formula_26, we might have formula_34. The adjoint representation, for example, is always isomorphic to its dual.