The type of the quadratic form

Question

I have been doing the following exercise, and there is one line that I do not understand.

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I have solved a) and got

Regarding b) I realised that

Now what I do not understand: After this it says in the solutions: I.e. after a rotation with

$$\begin{align}y_1&= x_1 + x_{2n}\\&\vdots\\ y_n &= x_n + x_{n+1}\\y_{n+1}& = x_1 − x_{2n}\\&\vdots\\ y_{2n}& = x_n − x_{n+1}\end{align}$$

we see that the quadratic form is indefinite, when it is written as:

My question is what does this part after a rotation with... mean? Thank you!