This is the homogeneous equation of a projective conic. The standard recipe for a rational parametrization of such curves is to
- Find at least one rational point on the curve (by, e.g., using linear coordinate transformations to reduce it to the Legendre form
$$A X^2 + BY^2 + C Z^2 =0$$
and using the Hasse local-global principle), then
- using stereographic projection from that point to produce the parametrization.
In the present case we can see that
$$[x : y : z]=[1 : -1 : 1]$$
is a solution; stereographic projection onto $y=0$, say, gives
$$[x : y : z] =
[ a^2-ab+b^2 : b^2 - ab - a^2 : a^2 - ab - b^2 ]\text{.}$$
Edit: it should be noted that parametrizing projective conics is routine enough to be relegated to a computer algebra system. For example, in Magma the input
k := Rationals();
P2<x,y,z> := ProjectiveSpace(k, 2);
f := x^2 - y^2 - z^2 - x*y - x*z - y*z;
C := Conic(P2, f);
P1<t,u> := ProjectiveSpace(k, 1);
Parametrization(C, Curve(P1));
gives
Mapping from: Curve over Rational Field defined by
0 to CrvCon: C
with equations :
t^2 + t*u + u^2
-t^2 + t*u + u^2
-t^2 - 3*t*u - u^2
and inverse
2*x - 2*y
-2*x - 2*z
and alternative inverse equations :
-2*x - 2*z
4*x + 2*y + 2*z
i.e.,
$$\begin{align}
[ x : y : z] &=
[ t^2 + tu + u^2 :
-t^2 + tu + u^2 :
-t^2 - 3tu - u^2 ] \\
[t : u ] &= [ 2x - 2y : -2x - 2z ]
\end{align}$$