  The Bannai-Ito algebra $BI(n)$ is viewed as the centralizer of the action of
$\mathfrak{osp}(1|2)$ in the $n$-fold tensor product of the universal algebra
of this Lie superalgebra. The generators of this centralizer are constructed
with the help of the universal $R$-matrix of $\mathfrak{osp}(1|2)$. The
specific structure of the $\mathfrak{osp}(1|2)$ embeddings to which the
centralizing elements are attached as Casimir elements is explained. With the
generators defined, the structure relations of $BI(n)$ are derived from those
of $BI(3)$ by repeated action of the coproduct and using properties of the
$R$-matrix and of the generators of the symmetric group $\mathfrak S_n$.
