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The equations of the Cayley surface and of the dual surface:
 
    The approach used here is an extension of what I have learned from Sevín Recillas.  Some choices of emphasis are recent, but I have stayed fairly close to methods that Sevín and I discussed in Morelia.

    In the previous discussion, we obtained the following version of the implicit equation of the dual surface :
 
      64wxyz = (w² + x² + y² + z² - 2wx - 2wy - 2wz - 2xy - 2xz - 2yz)².
 
As I noted in a previous page, this equation (i) is symmetric in the variables w,x,y,z and (ii) clearly exhibits fact that each of the coordinate planes intersects the dual surface (with multiplicity = 2) in a plane conic curve. 

    In this page I will present a change of coordinates that yields a version of the implicit equation which exhibits the singular locus much more directly.  I will initially use a set of variables which is not completely symmetric:
 
      a = w + x     b = w - x;     c = y + z;     d = y - z.
 
As linear combinatons of  w,x,y,z  the variables  a,b,c,d  are linearly independent.  {We are not considering fields of characteristic 2.}  With these variables, we have  4wx = a² - b²  and   4yz = c² - d²,  while:
 
      w² + x² + y² + z² - 2wx - 2wy - 2wz - 2xy - 2xz - 2yz = b² +d² -2ac.
 
Therefore, we have the following version of the implicit equation:
 
      4(a² - b²)(c² - d²) = (b² + d² -2ac)²,
or:
      4a²c² + 4b²d² - 4a²d² - 4b²c² = (b² + d²)² - 4ac(b² + d²) + 4a²c².
 
One obvious cancellation is possible here; if we do this cancellation and also subtract  4b²d²  from both sides of the equation, then we obtain a fairly short version of the implicit equation:
 
      - 4a²d² - 4b²c² = (b² - d²)² - 4ac(b² + d²). 

    Now, the plane  b + d = 0  (or equivalently  w - x + y - z = 0)  contains two of the three lines that compose the singular locus of the dual surface, specifically  w=x,  y=z  and  w=z,  x=y.  Similarly, the plane  b - d = 0  (or equivalently  w - x - y + z = 0)  contains the lines w=x,  y=z  and  w=y,  x=z.  This suggests that  b and d are appropriate variables to use in a coordinate system that relates naturally to the singular locus of the dual surface.  Indeed, the first term on the right side of our implicit equation is equal to  (b + d)²(b - d)². 

    Another variable with a similar property is  a - c = w + x - y - z.  We can introduce this variable into the equation by adding  2(a² + c²)(b² + d²)  to both sides of the implicit equation, as follows:
 
      2(a² - c²)(b² - d²) = (b² - d²)² + 2(a - c)²(b² + d²), 
or:
      2(a² - c²)(b² - d²) = (b + d)²(b - d)² + (a - c)²(b + d)² + (a - c)²(b - d)². 

    Our final change of variables can now be introduced:

• W = a + c = w + x + y + z;
• X = a - c = w + x - y - z;
• Y = b + d = w - x + y - z;
• Z = b - d = w - x - y + z.

    We also should note that the variables  W,X,Y,Z  are linearly independent as linear combinations of  a,b,c,d  and therefore also as linear combinations of  w,x,y,z.  Alternatively, the change of variables from  w,x,y,z  to  W,X,Y,Z  is accomplished (in either direction up to a scalar multiple) by the following matrix:
 
           1     1      1     1
           1     1     -1    -1
           1    -1      1    -1
           1    -1     -1     1
 
Indeed, if we multiply this matrix by ½, we obtain a symmetric orthogonal matrix.