  The recent interest in the study of higher-rank polynomial algebras related
to $n$-dimensional classical and quantum superintegrable systems with coalgebra
symmetry and their connection with the generalised Racah algebra $R(n)$, a
higher-rank generalisation of the rank one Racah algebra $R(3)$, raises the
problem of understanding the role played by the $n - 2$ quadratic subalgebras
generated by the left and right Casimir invariants (sometimes referred as
universal quadratic substructures) from this new perspective. Such subalgebra
structures play a signficant role in the algebraic derivation of spectrum of
quantum superintegrable systems. In this work, we tackle this problem and show
that the above quadratic subalgebra structures can be understood, at a fixed $n
> 3$, as the images of $n - 2$ injective morphisms of $R(3)$ into $R(n)$. We
show that each of the $n-2$ quadratic subalgebras is isomorphic to the rank one
Racah algebra $R(3)$. As a byproduct, we also obtain an equivalent presentation
for the universal quadratic subtructures generated by the partial Casimir
invariants of the coalgebra. The construction, which relies on explicit
(symplectic or differential) realisations of the generators, is performed in
both the classical and the quantum cases.
