### B.2 of signature

When is Lorentzian, there also exists a natural embedding of the Clifford algebra
into depending on a sign :
This embedding induces an embedding , . Moreover if is odd,
one has
Plugging this into the expression for , we see that a -parallel spinor in the cone, restricts to
to a geometric Killing spinor obeying
Furthermore,
- if is even: there exists a one-to-one correspondence between parallel spinors in
and geometric Killing spinors in ; and
- if is odd: there exists a one-to-one correspondence between parallel spinors in
with definite chirality
and geometric Killing spinors in .