Symmetric subalgebras noncohomologous to zero in a complex semisimple Lie algebra and restrictions of symmetric invariants

Let be a semisimple involutory Lie algebra and the corresponding symmetric pair. Let be a fundamental Cartan subalgebra of containing a Cartan subalgebra of . The semisimple involutory Lie algebras with the symmetric subalgebra noncohomologous to zero in are completely classified by showing that is noncohomologous to zero in if and only if the Spinν representation of is primary. Based on this result we then determine the image of the restriction map , where and are the respective Weyl groups.