### 4.3 The Chevalley involution

The symmetry between the positive and negative subalgebras and of the Kac–Moody algebra
can be rephrased formally as follows: The Kac–Moody algebra is invariant under the Chevalley involution
, defined on the generators as
The Chevalley involution is in fact an algebra automorphism that exchanges the positive and negative sides
of the algebra.
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 [116], 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
ideals
(see [116] and Section 4.5).