  Let ${\mathfrak g}$ denote the complexified Lie algebra of $G={\mathrm
O}(p,q)$ and $K$ a maximal compact subgroup of $G$. In the previous paper, we
constructed $({\mathfrak g},K)$-modules associated to the finite-dimensional
representation of ${\mathfrak sl}_2$ of dimension $m+1$, which we denote by
$M^{+}(m)$ and $M^{-}(m)$. The aim of this paper is to show that the
annihilator of $M^{\pm}(m)$ is the Joseph ideal if and only if $m=0$. We shall
see that an element of the symmetric of square $S^{2}({\mathfrak g})$ that is
given in terms of the Casimir elements of ${\mathfrak g}$ and the complexified
Lie algebra of $K$ plays a critical role in the proof of the main result.
