Null structure groups in eleven dimensions

We classify all the structure groups which arise as subgroups of the isotropy group, (Spin(7) ⋉ R8) × R, of a single null Killing spinor in eleven dimensions. We construct the spaces of spinors fixed by these groups. We determine the conditions under which structure subgroups of the maximal null strucuture group (Spin(7) ⋉ R8) × R may also be embedded in SU(5), and hence the conditions under which a supersymmetric spacetime admits only null, or both timelike and null, Killing spinors. This concludes, for spacetimes admitting at least one null Killing spinor, the first, purely algebraic, phase of the refined G-structure classification of supersymmetric spacetimes in eleven dimensions, the objective of which is the derivation of the general local bosonic solution of the Killing spinor equation of eleven dimensional supergravity. Finally, we discuss how the second phase of the classification, involving the direct analysis of the Killing spinor equation, may be most efficiently organised.