  We study the algebraic structure of the Killing superalgebra of a
supersymmetric background of $11$-dimensional supergravity and show that it is
isomorphic to a filtered deformation of a $\mathbb Z$-graded subalgebra of the
Poincar\'e superalgebra. We are able to map the classification problem for
highly supersymmetric backgrounds (i.e., those which preserve more than half
the supersymmetry) to the classification problem of a certain class of filtered
deformations of graded subalgebras of the Poincar\'e superalgebra. We show that
one can reconstruct a highly supersymmetric background from its Killing
superalgebra; in so doing, we relate the bosonic field equations of
$11$-dimensional supergravity to the Jacobi identity of the Killing
superalgebra and show in this way that preserving more than half the
supersymmetry implies the bosonic field equations.
