Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

The problem of providing complete presentations reduction algebras associated to a pair Lie $$( \mathfrak{G} , \mathfrak{g} )$$ has previously been considered by Khoroshkin and Ogievetsky in the case diagonal algebra for $$\mathfrak{gl}(n)$$ . In this paper, we consider superalgebras \mathfrak{osp}(1|2) \times as double coset space having an associative $$ \mathrel{\scriptstyle\lozenge} -product give presentation terms generators relations. We also provide PBW basis along with Casimir-like elements subgroup automorphisms.