Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra

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edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.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.

• You might check out Lectures 26 and 27 in these notes: stacky.net/files/written/LieGroups/LieGroups.pdf These are lectures of Borcherds, who bases the construction of $E_8$ on constructions of Kac-Moody algebras from vertex algebras. So this might be a good warm-up. – Ian Agol Apr 19 '14 at 4:58

• This is on page 222 in Mathieu's monograph, in XVI $\S4$, in case the numbering system isn't immediately clear. – Jim Humphreys Oct 13 '14 at 23:02