In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.


Does anyone know how this proof goes?


For reference, 9.11 is the statement that given a symmetrizable generalized Cartan matrix $A$, with associated lie algebra $\tilde{\mathfrak{g}}(A)$ with only the Weyl relations, the maximal ideal not intersecting $\mathfrak{h}$ is generated by the Serre relations. Thank you in advance. 



Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra
The proof is written in 


* Mathieu, Olivier. Formules de caractères pour les algèbres de Kac-Moody générales. (French) [Character formulas for general Kac-Moody algebras] Astérisque No. 159-160 (1988), 267 pp. [MR0980506 (90d:17024)](http://www.ams.org/mathscinet-getitem?mr=980506)


It uses results of Garland-Lepowsky and Garland-Kac. See Lemma 118. (I have to thank Ivan Angiono for the reference.)


