  We perform a unified systematic analysis of $d+1$ dimensional, spin $\ell$
representations of the isometry algebra of the maximally symmetric spacetimes
AdS$_{d+1}$, $\mathbb{R}_{1,d}$ and dS$_{d+1}$. This allows us to explicitly
construct the effective low-energy bulk equations of motion obeyed by linear
fields, as the eigenvalue equation for the quadratic Casimir differential
operator. We show that the bulk description of a conformal family is given by
the Fierz-Pauli system of equations. For $\ell = 2$ this is a massive gravity
theory, while for $\ell = 2$ conserved currents we obtain Einstein gravity and
covariant gauge fixing conditions. This analysis provides a direct algebraic
derivation of the familiar AdS holographic dictionary at low energies, with
analogous results for Minkowski and de Sitter spacetimes.
