  Let $\mathfrak{co}(J)$ be the conformal algebra of a simple Euclidean Jordan
algebra $J$. We show that a (non-trivial) unitary highest weight
$\mathfrak{co}(J)$-module has the smallest positive Gelfand-Kirillov dimension
if and only if a certain quadratic relation is satisfied in the universal
enveloping algebra $U(\mathfrak{co}(J)_{\mathbb{C}})$. In particular, we find
an quadratic element in $U(\mathfrak{co}(J)_{\mathbb{C}})$. A prime ideal in
$U(\mathfrak{co}(J)_{\mathbb{C}})$ equals the Joseph ideal if and only if it
contains this quadratic element.
