Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix. Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$. Write $G_2$ for the subset of elements of order dividing 2 in $G$.
Question. Is it true that the inclusion $H_2\hookrightarrow G_2$ induces a bijection between $H_2/W$ and the set of $G$-conjugacy classes in $G_2$? If not, is it true at least for affine Kac-Moody groups?
