  We study the smooth projective symmetric variety of Picard number one that
compactifies the exceptional complex Lie group G2, by describing it in terms of
vector bundles on the spinor variety of Spin(14). We call it the double Cayley
Grassmannian because quite remarkably, it exhibits very similar properties to
those of the Cayley Grassmannian (the other symmetric variety of type G2), but
doubled in the certain sense. We deduce among other things that all smooth
projective symmetric varieties of Picard number one are infinitesimally rigid.
