Nilpotence (x^2 = 0) is also very helpful some times: it's a process which is self-reversing. Like the discrete Fourier transform (if you set up the constants properly).
In mathematics, a self-reversing function is called an involution, and it's f^2 (or f(f) ) = Id, the identity function.
Nilpotence is very different. It says that if you apply your function a certain number of times, you end up with zero no matter what the input is. For example, projection on x axis + 90 ° rotation of a vector is nilpotent.
f(x) = f(f(x)) = f f x = f^2 x = f x
And leaving out the x, because it is just a placeholder anyways:
f f = f^2 = f
And of course this means that
f^n = f because f^n-1 f = f^n-1 by induction.
Particularly in this case O^2 = O makes more sense to me than f(f(x)) = f(x). And I just naturally think about A B - B A = 0 rather than f(g(x)) - g(f(x)) = 0 for commutativity.
Also you may be confusing it with x^n = 1 (which I'm not sure how to name, 'root of unity' perhaps). This would be the case for the Fourier transform (with n=4).
If x^2 = 0 then applying the Fourier transform twice would null your function, which isn't the case.
Squaring a upper triangular matrix with 0 on the diagonal is nilpotent. Derivatiting a polynomial of degree N is nilpotent after N iteration.