  A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$
on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a
consequence one obtains the Laplace-Runge-Lenz vector for each classical
$\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations
is greatly simplified via an idea due to A. Weinstein. The totality of these
Poisson realizations is shown to be equivalent to the canonical Poisson
realization of $\mathfrak {so}^*(4n)$ on the Poisson manifold $T^*\mathbb
H_*^n/\mathrm{Sp}(1)$. (Here $\mathbb H_*^n:=\mathbb H^n\backslash \{0\}$ and
the Hamiltonian action of $\mathrm{Sp}(1)$ on $T^*\mathbb H_*^n$ is induced
from the natural right action of $\mathrm{Sp}(1)$ on $\mathbb H_*^n$. )
