Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero

Assume that $\mathbb F$ is an algebraically closed with characteristic $0$. The Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative $\mathbb F$-algebra generated by $X,Y,Z$ and the relations assert that each of \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} is central in $\mathfrak{BI}$. In this paper we classify the finite-dimensional irreducible $\mathfrak{BI}$-modules up to isomorphism. As we will see the elements $X,Y,Z$ are not always diagonalizable on finite-dimensional irreducible $\mathfrak{BI}$-modules.