Indice et formes linéaires stables dans les algèbres de Lie

We characterize, in a purely algebraic manner, certain linear forms, called stable, introduced in [Y. Kosmann, S. Sternberg, Conjugaison des sous-algèbres dʹisotropie, C. R. Acad. Sci. Paris Ser. A 279 (1974) 777–779], on a Lie algebra. As an application, we determine the index of a Borel subalgebra of a semi-simple Lie algebra. Finally, we give an example of a parabolic subalgebra of a semi-simple Lie algebra which does not admit any stable linear form.