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Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]


Have I missed something - my apologies if this is answered in a FAQ
I want to make the simple Re and Im parts simplify properly?

test = 
  {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}

FullSimplify[test, eta > 1]
gives*{Sqrt[-1 + eta^2], 0}

But
FullSimplify[test, eta < 1]
gives
{Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}

Needs["Algebra`ReIm`"] does not seem to help

Real numbers demonstrate what should happen:
test) /. {{eta -> 0.1}, {eta -> 2}}
{{0., 0.99498743710662}, {Sqrt[3], 0}}


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