Casimir operators induced by Maurer-Cartan equations

It is shown that for inhomogeneous Lie algebras $\mathfrak{g}=\mathfrak{s}{\overrightarrow{\oplus }}_{\mathrm{\Lambda }}(\dim \mathrm{\Lambda })L_1$ satisfying the condition $\mathcal{N}(\mathfrak{g})=1$, the only Casimir operator can be explicitly constructed from the Maurer-Cartan equations by means of wedge products. It is shown that this constraint imposes sharp bounds for the dimension of the representation $R$. The procedure is generalized to compute also the rational invariant of some Lie algebras.