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

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