  Given a linear representation $\rho : \mathfrak{g} \longrightarrow
\mathfrak{g}\ell(V)$ of a Lie algebra $\mathfrak{g}$, one can define a linear
representation $\rho_m : \mathfrak{g}_m \longrightarrow \mathfrak{g}\ell(V^m)$
of the generalized Takiff algebra $\mathfrak{g}_m$. It is proved here that the
vector fields defined by $\rho_m$ on $V^m$ do have the Dixmier property if
those defined by $\rho$ have the same property. Examples where the result
applies are given and in particular, those of the adjoint or coadjoint
representations of Takiff algebras.
