The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g we study the space of g-invariants in the tensor product of the exterior algebra of g^* with g^*, which describes the isotypic component of type g in the exterior algebra, as a module over the algebra of invariants. As main result we prove that A is a free module, of rank twice the rank of g, over the exterior algebra generated by all primitive invariants with the exception of the one of highest degree.