Doesn't change much in this particular instance, but you can use Refine
to incorporate assumptions on the parameters in your equations. Keep in mind that expressions in inequality assumptions are assumed to be real.
eqns = Refine[{1 == -x + y + z,
0 == y (-4 a/(b - Sqrt[b^2 + 8 a^2])) + z (-4 a/(b + Sqrt[b^2 + 8 a^2])),
0 == x + y + z},
Assumptions -> Element[x, Reals] && Element[y, Reals]]
(* {1 == -x + y + z,
0 == -((4 a y)/(b - Sqrt[8 a^2 + b^2])) - (4 a z)/(b + Sqrt[8 a^2 + b^2]),
0 == x + y + z} *)
Solve[eqns, {x, y, z}]
(* {{x -> -(1/2),
y -> -((b - Sqrt[8 a^2 + b^2])/(4 Sqrt[8 a^2 + b^2])),
z -> -((-b - Sqrt[8 a^2 + b^2])/(4 Sqrt[8 a^2 + b^2]))}} *)
Reduce[eqns, {x, y, z}]
(* 8 a^2 + b^2 != 0 && a != 0 && x == -(1/2) &&
y == (8 a^2 + b^2 - b Sqrt[8 a^2 + b^2])/(4 (8 a^2 + b^2)) &&
z == (8 a^2 + b^2 + b Sqrt[8 a^2 + b^2])/(4 (8 a^2 + b^2)) *)