  We recover the classification of the maximally supersymmetric bosonic
backgrounds of eleven-dimensional supergravity by Lie algebraic means. We
classify all filtered deformations of the $\mathbb Z$-graded subalgebras
$\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$
of the Poincar\'e superalgebra
$\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}=V\oplus
S\oplus \mathfrak{so}(V)$ which differ only in zero degree, that is
$\mathfrak{h}_0\subset\mathfrak{g}_0$ and $\mathfrak{h}_j=\mathfrak{g}_j$ for
$j<0$. Aside from the Poincar\'e superalgebra itself and its $\mathbb Z$-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.
