But the geometric product (a' /\ a")(a" /\ a') equals -1 . (No?)
It had better be (No).
Write the wedge products in terms of the geometric product:
P /\ Q = (P Q - Q P)/2
for any vectors P and Q.
So in this case we are interested in
(a'a" - a"a')(a"a' - a'a")/4
which expands as
(a'a"a"a' - a'a"a'a" - a"a'a"a' + a"a'a'a")/4
which equals (1+1+1+1)/4 since a' and a" are orthogonal spacelike
unit vectors.
a'a" = -a"a' orthogonal
a'a' = a"a" = +1 spacelike unit vectors