The general expresion for composition of velocities is

a*b = sqrt((a^2 + b^2 + 2ab cos(alpha)) -
          (ab sin(alpha))^2/c^2 ) /(1 + ab cos(alpha)/c^2),
where alpha is the angle between the velocities a and b.

If alpha=0, then the binary operator * reduces to
    a*b = (a + b) /(1 + ab/c^2)

if alpha = pi/2, then
   a*b = sqrt(a^2 + b^2 - (ab)^2/c^2)

if alpha = pi, then
   a*b = (a - b) /(1 - ab/c^2)

Since the operator * is defined as a function of alpha, we see it is
not unique.