# Simplify root inside of root

xxx=(2 Sqrt[1 - 2/(1 + Sqrt[G1])] G1 +
2 Sqrt[((-1 + Sqrt[G1]) G1)/(1 + Sqrt[G1])])/(-1 + 2 G1)


I have quite complicated expression. I want to simplify little better. I tried

FullSimplify[xxx, TransformationFunctions -> {Automatic, PowerExpand},
Assumptions -> {G1 > 1}]


You can see that is can be simplified to by hand

(2 Sqrt[-1 + G1] Sqrt[G1])/(-1 + 2 G1)


however, it does not work in this case. Any suggestion?

Weirdly, the less specific assumption $$G1>0$$ works to achieve your simplification here:
Simplify[xxx, Assumptions -> G1 > 0]

FullSimplify[xxx /. G1 -> g2^2, Assumptions -> g2 > 1] /.