  The analysis of the $\mathbb{Z}_2^{3}$ Laplace-Dunkl equation on the
$2$-sphere is cast in the framework of the Racah problem for the Hopf algebra
$sl_{-1}(2)$. The related Dunkl-Laplace operator is shown to correspond to a
quadratic expression in the total Casimir operator of the tensor product of
three irreducible $sl_{-1}(2)$-modules. The operators commuting with the Dunkl
Laplacian are seen to coincide with the intermediate Casimir operators and to
realize a central extension of the Bannai-Ito (BI) algebra. Functions on $S^2$
spanning irreducible modules of the BI algebra are constructed and given
explicitly in terms of Jacobi polynomials. The BI polynomials occur as
expansion coefficients between two such bases composed of functions separated
in different coordinate systems.
