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Is is true that if A is real with A²=I (eigenvalues ±1), it is diagonalizable overR?

What if I add that A is in O(m,m), where O(m,m) is the split indefinite orthogonal group of 2m x 2m matrices M such that [itex]M^TI_{m,m}M=I_{m,m}[/itex], where [itex]I_{m,m}[/itex] is the block diagonal matrix diag(I_m,-I_m) for I_m the m x m identity matrix?

Thanks

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# A²=Id => A diagonalizable?

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