Damn irrationals

You have the 3 numbers {4, 1+22, 2}\big\{4,\ 1+2\sqrt{2},\ \sqrt{2}\big\} on a blackboard. You are permitted to perform the following operation whenever you want and as many times as you want:

  • You can choose 2 numbers aa and bb and replace them with a+b2\frac{a+b}{\sqrt{2}} and ab2\frac{a-b}{\sqrt{2}}.

Is it possible to attain the 3 numbers {3+2, 22, 21}\big\{3+\sqrt{2},\ 2\sqrt{2},\ \sqrt{2}-1\big\} after a finite number of operations?

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