In this Letter, we present some comments on the nature of the CCGSD ansatz psi = e(T) Phi for an n-electron wave function, with T a linear combination of general one- and two-particle operators. Nooijen had conjectured that such a parameterization is possible for the exact eigenstate psi. We point out that the essential reason for the invalidity of this conjecture lies in the fact that the basis operators into which the Fock space Hamiltonian can be expanded, are not closed under commutation, i.e. do not span a Lie algebra. We give two proofs, based, respectively, on the variation principle and the method of moments, to show this. The variational proof traces the same ground as in an earlier one by Nakatsuji, which did not get the due recognition. We, however, formulate it in such a way that the key role of the algebra of the operators of T becomes transparent. We also discuss a related proof of Mazziotti. Our second proof sheds further light on why the ansatz, while not exact, might still provide an accurate description of a many-electron state, as has been found in some recent computational studies. We also discuss two recent attempts to disprove the exactness of the ansatz, based on a dimensionality argument. (C) 2004 Elsevier B.V. All rights reserved.