Cross Products Around the Octagon

There are a total of $8\choose2$ $= 28$ pairs of unit vectors where $i < j$ : 12 pairs make an angle of $45^\circ$ or $135^\circ$ , 6 pairs make an angle of $90^\circ$ , 4 pairs make an angle of $180^\circ$ , 4 pairs make an angle of $225^\circ$ or $315^\circ$ , and finally only 2 pairs make an angle of $270^\circ$ .

The cross product of those vectors that form a $45^\circ$ or $135^\circ$ angle is $\langle 0, 0, \frac{\sqrt{2}}{2} \rangle$ .
The cross product of those vectors that form a $90^\circ$ angle is $\langle 0, 0, 1\rangle$ .
The cross product of those vectors that form a $180^\circ$ angle is $\langle 0, 0, 0\rangle$ .
The cross product of those vectors that form a $225^\circ$ or $315^\circ$ angle is $\langle 0, 0, -\frac{\sqrt{2}}{2} \rangle$ .
The cross product of those vectors that form a $270^\circ$ angle is $\langle 0, 0, -1\rangle$ .

Adding all these cross products up gives:

$12 \langle 0, 0, \frac{\sqrt{2}}{2} \rangle + 6 \langle 0, 0, 1 \rangle + 4\langle 0, 0, -\frac{\sqrt{2}}{2} \rangle + 2\langle 0, 0, -1 \rangle = \langle 0, 0, 4\sqrt{2} + 4\rangle$