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The Racah algebra of rank $n$ can be constructed within the $n+2$-fold tensor product of the universal enveloping algebra of $su(1,1)$. So far two concrete realizations of this construction have received quite a bit of attention. I'll summarize them, and will also construct a third, new realization. On top of its simplicity, it has the advantage of yielding an $n$-variable realization of the rank $n$ case. This is joint work with P. Iliev and L. Vinet.