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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 ZZ -graded subalgebras h=h−2⊕h−1⊕h0h=h−2⊕h−1⊕h0 of the Poincaré superalgebra g=g−2⊕g−1⊕g0=V⊕S⊕so(V)g=g−2⊕g−1⊕g0=V⊕S⊕so(V) which differ only in zero degree, that is h0⊂g0h0⊂g0 and hj=gjhj=gj for j<0j<0 . Aside from the Poincaré superalgebra itself and its ZZ -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.
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