Octonionic D=11 Supergravity and Octavian Integers as Dilaton Vectors

We formulate D=11 supergravity over the octonions by rewriting 32-component Majorana spinors as 4-component octonionic spinors. Dimensional reduction to D=4 and D=3 suggests an interpretation of the so-called dilaton vectors, which parameterise the couplings of the dilatons to other fields in the theory, as unit octavian integers - the octonionic analogues of integers. The parameterisation involves a novel use of the duality between points and lines on the Fano plane, and suggests a series of consistent truncations with N=8,4,2,1, giving the four curious supergravities studied by Duff and Ferrara