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

La propri\'et\'e de Dixmier pour les alg\`ebres de Lie de champs de vecteurs