  We consider geometric and supergravity Killing spinors and the spinor
bilinears constructed out of them. The spinor bilinears of geometric Killing
spinors correspond to the antisymmetric generalizations of Killing vector
fields which are called Killing-Yano forms. They constitute a Lie superalgebra
structure in constant curvature spacetimes. We show that the Dirac currents of
geometric Killing spinors satisfy a Lie algebra structure up to a condition on
2-form spinor bilinears. We propose that the spinor bilinears of supergravity
Killing spinors give way to different generalizations of Killing vector fields
to higher degree forms. It is also shown that those supergravity Killing forms
constitute a Lie algebra structure in six and ten dimensional cases. For five
and eleven dimensional cases, the Lie algebra structure depends on an extra
condition on supergravity Killing forms.
