One of the easiest and most straightforward ways of defining twistors uses the transformation properties of linear and angular momentum of a particle under a shift of origin.
Consider a change of origin from 0 to a point Q with coordinates 
.
With respect to the new origin,
We define the Pauli-Lubanski spin vector:
It is easy to show that 
.
Now let 
be future null, so that we can write 
.
Since 
is skew it can be decomposed as
The dual is then easy to write:
and
In nature we only observe massles particles with definite handedness, i.e. with 
.
follows immediately and hence 
, where either 
or 
are proportional to 
(note that 
always) and 
denotes index symmetrisation.
The same arguments apply to 
.
We can now define 
with
At last, we can say, 
is a twistor.
Now we have
if we define 
.