Casimir operators induced by the Maurer-Cartan equations

It is shown that for inhomogeneous Lie algebras with only one Casimir operator, the latter can be explicitly constructed from the Maurer-Cartan equations by means of wedge products. It is further proved that this constraint imposes sharp bounds for the dimension of the representation R defining the semidirect product. The procedure is generalized to compute also the rational invariant of some Lie algebras.