Refining G-structure classifications
  Using G-structure language, a systematic, iterative formalism for computing neccessary and sufficient conditions for the existence of N arbitrary linearly independent Killing spinors is presented. The key organisational tool is the common isotropy group of the Killing spinors. The formalism is illustrated for configurations in gauged SU(2) supergravity in seven dimensions admitting at least one null Killing spinor, and the possible isotropy groups are shown to be $(SU(2)\ltimes\mathbb{R}^4)\times\mathbb{R}$, SU(2), $\mathbb{R}^5$, or the identity. The constraints associated with the existence of certain additional Killing spinors are computed, and used to derive numerous solutions. A discussion of the relevance of the formalism to the complete classification of all supersymmetric configurations in d=11 is given. 
Oisin A. P. Mac Conamhna