Spencer cohomology and eleven-dimensional supergravity

We recover the classification of the maximally supersymmetric bosonic backgrounds of 11-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the Z-graded subalgebras h=h_{−2}+h_{−1}+h_{0} of the Poincaré superalgebra g=g_{−2}+g_{−1}+g_{0}=V+S+so(V) which differ only in zero degree, that is h_{0}⊂g_{0} and h_{j}=g_{j} for j<0. Aside from the Poincaré superalgebra itself and its Z-graded subalgebras, there are only three other Lie superalgebras, which are the symmetry superalgebras of the non-flat maximally supersymmetric backgrounds. In passing we identify the gravitino variation with (a component of) a Spencer cocycle.