What is the group-like structure on $x^2+y^2+z^2-2xyz=1$? (Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.) 
Solving for $ z $ gives
$$
z=xy \pm \sqrt{(1-x^2)(1-y^2)},
$$
where $-1\leq x,y \leq 1$. Now choose the positive square root and call it $ x\oplus y$.
(Alternatively, we can define $x\oplus y=\cos(\arccos(x)+\arccos(y))$.)
We can check commutativity, identity ($1$), inverses ($-x$). Associativity is might be a bit harder, but I reckon it works.
The defining equation is a cubic, hence, it reminds me of the group structure of an elliptic curve, but an elliptic curve has two variables and the group law is defined by intersection with a line, while this equation has three variable, one is the "sum" of the other two.
Question
Has anyone seen a structure like this before?
 A: This is better understood by considering the roots of unity. Suppose
$\, u_n = r_n + s_n\sqrt{-1} \,$ for $\, n=1,2,3 \,$ be three roots of unity such that $\, u_3 = u_1 u_2. \,$ Then 
 $\, r_1^2 + r_2^2 + r_3^2 - 2 r_1 r_2 r_3 = 1. \,$ The same equation is true if $\, u_3 = u_1/u_2 \,$ or $\, u_3 = u_2/u_1. \,$ Or more symmetrically if $\, u_1u_2u_3 = 1.\,$ The reason is that given only the real part of a root of unity, there are two possible values for the imaginary part. The roots of unity form a multiplicative group, but if we only have the real part, the group operation is two-valued. 
The case of addition of points on elliptic curves is very appropriate. The group operation there is related to three points on a line, but not in the obvious way. That is, the third point is not the sum of the other two, but rather the negative of the sum. Another way to state it is that the sum of all three points is the identity. Also, given only the $\,x\,$ coordinate of a point, there are two values for the $\,y\,$ coordinate. There is some similarity between  "adding" points on the unit circle and points on an elliptic curve.
A: At first glance it seems to be the same as Cayley's nodal cubic surface. Although there it is described by the (projective) equation $$wxy+xyz+wxz+wyz=0.$$ How, then, do obtain my version of it (if, in fact, they are the same.)
Here is a page which seems to derive my equation from Cayley's.
A: You also can have
• $(2,3,6\pm\sqrt{24})$ so all three are greater than 1.
• $(-2,-3,6\pm\sqrt{24}$
• $(1,y,y)$
• $(-1,y,-y)$
Remember $\cos(a+b)=\cos a\cos b -\sin a\sin b$ so the choice of square-root might get tricky.
