  We consider a second order operator with analytic coefficients whose
principal symbol vanishes exactly to order two on a symplectic real analytic
manifold. We assume that the first (non degenerate) eigenvalue vanishes on a
symplectic submanifold of the characteristic manifold. In the $C^\infty$
framework this situation would mean a loss of 3/2 derivatives. We prove that
this operator is analytic hypoelliptic. The main tool is the FBI transform. A
case in which $C^\infty$ hypoellipticity fails is also discussed.
