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
.