  In this paper we clarify the relations occurring among the osp(1|32) algebra,
the M-algebra and the hidden superalgebra underlying the Free Differential
Algebra of D=11 supergravity (to which we will refer as DF-algebra) that was
introduced in the literature by D'Auria and Fr\'e in 1981 and is actually a
(Lorentz valued) central extension of the M-algebra including a nilpotent
spinor generator, Q'. We focus in particular on the 4-form cohomology in 11D
superspace of the supergravity theory, strictly related to the presence in the
theory of a 3-form $A^{(3)}$. Once formulated in terms of its hidden
superalgebra of 1-forms, we find that $A^{(3)}$ can be decomposed into the sum
of two parts having different group-theoretical meaning: One of them allows to
reproduce the FDA of the 11D Supergravity due to non-trivial contributions to
the 4-form cohomology in superspace, while the second one does not contribute
to the 4-form cohomology, being a closed 3-form in the vacuum, defining however
a one parameter family of trilinear forms invariant under a symmetry algebra
related to osp(1|32) by redefining the spin connection and adding a new
Maurer-Cartan equation. We further discuss about the crucial role played by the
1-form spinor $\eta$ (dual to the nilpotent generator Q') for the 4-form
cohomology of the eleven dimensional theory on superspace.
