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I have an intuition that at least one of the element should be in the form of diagonal matices with diagonal entries being 1 or -1 (e.g I3)

I don't know if there's any other possiblity

Please give a strong proof

(a similar question is that show the centre of the general linear group GL2(C) consisits of all scalar matrices, I can show all scalar matrices are in the centre, but how to show that they are the only kind?)

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    Your intuition must be wrong, since any matrix commutes with _itself_, and neither it nor itself are necessarily diagonal.2012-12-28
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    What is a "strong proof"?2012-12-28
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    My original statement below missed a case - when two elements of $SO(3)$ have orthogonal axes of rotation, and the are both $180$ degree rotations around their respective axes, then they also commute.2012-12-28
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    It is not clear to me why this question got down voted. Hence I will +1 it to compensate the down vote.2012-12-28

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