  This paper is threefold. First, we provide a new "universal" definition for
the Racah algebra of rank-2 as an extension of the rank-1 Racah algebra where
the generators are indexed by subsets and any three disjoint indexing sets
define a subalgebra isomorphic to the rank-1 case. With this definition, we
explore some of the properties of the algebra including verifying that these
natural assumptions are equivalent to other defining relations in the
literature. Second, we look at the symmetries of the generators of the rank-2
Racah algebra. Those symmetries allows us to partially make abstraction of the
choice of the generators and write relations and properties in a different
format. Last, we provide a novel representation of the Racah algebra. This new
representation requires only one generator to be diagonal and is based on an
expansion of the split basis representation from the rank-1 Racah algebra.
