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Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimpleLie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we constructpolynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of$\frak{g}$ that commute with $\frak{r}$ and transform like the generators of$\frak{s}$, up to a functional factor that turns out to be a Casimir operatorof $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of$\frak{g}$ in closed form, using the classical formulae for the invariants of$\frak{s}$. The behavior of virtual copies with respect to contractions of Liealgebras is analyzed. Applications to the class of Hamilton algebras and theirinhomogeneous extensions are given.

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators