  We consider the effective action of the Heterotic Superstring to first order
in alpha prime and derive the necessary and sufficient conditions that a field
configuration has to satisfy in order to admit at least one Killing spinor
using the spinor bilinear method and making minimal coordinate and frame
choices. As a previous step in this derivation, we compute the complete spinor
bilinear algebra using the Fierz identities, obtaining as a by-product the
algebra satisfied by the Spin(7) structure contained in the bilinears in an
arbitrary basis. We find the relations existing between the left-hand-sides of
the bosonic equations of motion evaluated on supersymmetric field
configurations using the Killing Spinor Identities instead of the (far more
complicated) integrability conditions of the Killing Spinor Equations as it is
common in the literature. We show how to include the Kalb-Ramond's Bianchi
identity in the Killing Spinor Identities.
