From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism \(\sharp \) from the universal Racah algebra \(\Re \) into \(U(\mathfrak {sl}_2)\) . We use the Casimir elements of \(\Re \) to describe the kernel of \(\sharp \) . By pulling back via \(\sharp \) every \(U(\mathfrak {sl}_2)\) -module can be viewed as an \(\Re \) -module. We show that for any finite-dimensional \(U(\mathfrak {sl}_2)\) -module V, the \(\Re \) -module V is completely reducible and three generators of \(\Re \) act on every irreducible \(\Re \) -submodule of V as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube H(D, 2) and a decomposition of the second distance operator of H(D, 2) induced by Johnson graphs.
