Finally, we quote the following useful theorem.
Theorem: The Kac–Moody algebra defined by the relations (4.10, 4.11) is simple.
The proof may be found in Kac’ book , page 12.
We note that invertibility and indecomposability of the Cartan matrix are central ingredients in the proof. In particular, the theorem does not hold in the affine case, for which the Cartan matrix is degenerate and has non-trivial ideals10 (see  and Section 4.5).
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