  We realize all irreducible unitary representations of the group
$\mathrm{SO}_0(n+1,1)$ on explicit Hilbert spaces of vector-valued
$L^2$-functions on $\mathbb{R}^n\setminus\{0\}$. The key ingredient in our
construction is an explicit expression for the standard Knapp-Stein
intertwining operators between arbitrary principal series representations in
terms of the Euclidean Fourier transform on a maximal unipotent subgroup
isomorphic to $\mathbb{R}^n$.
  As an application, we describe the space of Whittaker vectors on all
irreducible Casselman-Wallach representations. Moreover, the new realizations
of the irreducible unitary representations immediately reveal their
decomposition into irreducible representations of a parabolic subgroup, thus
providing a simple proof of a recent result of Liu-Oshima-Yu.
