Eleven-dimensional supergravity and octonions

We analyse an aspect of spontaneous compactification of 11-dimensional simple supergravity on the 7-sphere: the geometric meaning of the pseudo-scalar modes on S₇. In particular, we present the geometrical interpretation of the (anti) self-dual equation and of its 35 solutions. This is deduced from the properties of Killing spinors and of the algebra of octonions.