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