  It is shown that for inhomogeneous Lie algebras
$\frak{g}=\frak{s}\overrightarrow{\oplus}_{\Lambda}(\dim \Lambda)L_{1}$
satisfying the condition $\mathcal{N}(\frak{g})=1$, 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.
