We study `constrained generalized Killing (s)pinors', which characterize
supersymmetric flux compactifications of supergravity theories. Using geometric
algebra techniques, we give conceptually clear and computationally effective
methods for translating supersymmetry conditions into differential and
algebraic constraints on collections of differential forms. In particular, we
give a synthetic description of Fierz identities, which are an important
ingredient of such problems. As an application, we show how our approach can be
used to efficiently recover results pertaining to N=1 compactifications of
M-theory on eight-manifolds.Comment: 70 page