[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the isometries of [math]\mathbb C^2[/math] of determinant [math]1[/math] are

[[math]] U=\begin{pmatrix}a&b\\ -\bar{b}&\bar{a}\end{pmatrix}\quad,\quad |a|^2+|b|^2=1 [[/math]]

then work out as well the general case, of arbitrary determinant.