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We provide an explicit isomorphism between a quotient of the Bannai--Itoalgebra and the Brauer algebra. We clarify also the connection with the actionof the Lie superalgebra osp(1|2) on the threefold tensor product of itsfundamental representation. Finally, a conjecture is proposed to describe thecentralizer of osp(1|2) acting on three copies of an arbitrary finiteirreducible representation in terms of a quotient of the Bannai-Ito algebra.

Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra