I rewrote my demonstration:

Let two self adjoint operators such that AB = BA, let an eigenvalue of A and a self space associate. Now let not null vector such that:

How AB = BA, then:

Logo is invariant by B. Then v is an eigenvector common on A and B, then exist such that:

How and are real roots of characteristic polynomials of A and B, then and are both not invertible, then:

Then v belong to orthonormal base of eigenvectors of A and B, therefore the base diagonalize A and B simultaneously.