After the usefulness of the maximally supersymmetric IIB plane wave, it was natural to search for other maximally supersymmetric solutions, even though we had no good reason to expect that any more existed. Indeed, George and I showed this in Maximally supersymmetric solutions to ten- and eleven-dimensional supergravities. The proof for IIB was much more complicated than for eleven-dimensional supergravity, which we had obtained more than a year earlier. One of the conditions for maximal supersymmetry involves an algebraic condition on the fluxes, which in eleven-dimensional supergravity, we identified with the Plücker relations governing the projective embedding of the grassmannian of planes. The analogous algebraic condition in the case of IIB was a novel relation which generalises both the Plücker relation and the Jacobi identity. The geometric interpretation of this relation was given in Plücker-type relations for orthogonal planes, which also contains a conjecture which might be of independent interest to algebraic geometers.
I continued thinking about maximally supersymmetric solutions of other supergravity theories and trying to understand them conceptually in terms of established geometric ingredients. It is unnecessary to look at any supergravity theory which can be obtained as Kaluza-Klein reduction from ten- or eleven-dimensional supergravities, which leaves the chiral supergravities in six dimensions. Together with Ali Chamseddine and Wafic Sabra, we interpreted the maximally supersymmetric solutions of these theories as certain Lie groups admitting bi-invariant lorentzian metrics. These results are contained in Supergravity vacua and lorentzian Lie groups.