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

For a simple complex Lie algebra gg we study the space of invariants A=(⋀g⁎⊗g⁎)gA=(⋀g⁎⊗g⁎)g, which describes the isotypic component of type gg in ⋀g⁎⋀g⁎, as a module over the algebra of invariants (⋀g⁎)g(⋀g⁎)g. As main result we prove that A   is a free module, of rank twice the rank of gg, over the exterior algebra generated by all primitive invariants in (⋀g⁎)g(⋀g⁎)g, with the exception of the one of highest degree.