Exponentiability of quadratic Hamiltonians

We consider a Lie algebra of quadratic infinite polynomials of creation and annihilation operators and find that those algebras provide central extensions of some Lie algebras of bounded operators. We prove that the set of quadratic infinite polynomials of creation and annihilation operators corresponding to the ball {S is an element of inv(C) (H) \ parallel toSparallel to less than or equal to 1/3} is exponentiable on a dense subspace Gamma(0)H of the Fock space GammaH. This is done simultaneously both in the symmetric and the anti-symmetric case.