  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.
