mathfrak {osp}(1,2) and generalized Bannai-Ito algebras

Generalizations of the (rank 1) Bannai-Ito algebra are obtained from a refinement of the grade involution of the Lie super algebra $\mathfrak{osp}(1,2)$. A hyperoctahedral extension is derived by using a realization of $\mathfrak{osp}(1,2)$ in terms of Dunkl operators associated to the Weyl group $B_3$.