  Let $G$ be a real reductive connected Lie group and $\sigma$ an involution of
$G$. Let $H$ denote the identity component of the group of fixed points of
$\sigma$, $\mathfrak g$ the Lie algebra of $G$ and $\mathfrak q$ the -1
eigenspace of $\sigma$ in $\mathfrak g$. The group $H$ acts naturally on
$\mathfrak q$ via the adjoint representation. Let $C^{\infty}(\mathfrak q)^H$
denote the algebra of $H$-invariant smooth functions on $\mathfrak q$, and
$\mathfrak X(\mathfrak q)^H$ the space of $H$-invariant smooth vetor fields on
$\mathfrak q$. Any vetor field $X\in \mathfrak X(\mathfrak q)^H$ defines
naturally a derivation $D_X$ of the algebra $C^{\infty}(\mathfrak q)^H$. We
prove that the image of the map $X\mapsto D_X$ is the set of derivations of the
algebra $C^{\infty}(\mathfrak q)^H$ preserving the ideal
$\it{\Phi}C^{\infty}(\mathfrak q)^H$ of $C^{\infty}(\mathfrak q)^H$, where
$\it{\Phi}$ is a discriminant function on $\mathfrak q$.
