  This note focuses on some properties and uses of filtered deformations in the
context of D=11 supergravity. We define the concept of abstract symbol and give
a strong version of the Reconstruction Theorem, namely a bijective
correspondence from the space of highly supersymmetric supergravity backgrounds
to the space of abstract symbols. We propose a general strategy to construct
highly supersymmetric supergravity backgrounds and present an example in
detail, which includes the computation of the ideal generated by the Killing
spinors of two known pp-wave backgrounds with N=24 supersymmetry. Finally, we
give an alternative proof, based on the isotropy algebra of a supergravity
background, of a classical supersymmetry gap result of Gran, Gutowski,
Papadopoulos and Roest.
