Finite-dimensional irreducible modules for the three-point $\mathfrak{sl}_2$ loop algebra

Recently Brian Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra $\boxtimes$ by generators and relations, and displayed an isomorphism from $\boxtimes$ to the three-point $sl_2$ loop algebra. In this paper we describe the finite-dimensional irreducible $\boxtimes$-modules from multiple points of view.