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

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