The Hadamard formula is easy to show. The full BCH formula is a ***** (I spent several hours yesterday trying to do it, but I didn't understand enough about Lie groups to get there). Anyway, start with this function:
f(s) = e^{sA} B e^{-sA}
Then differentiate it a few times with respect to s:
f'(s) = e^{sA} A B e^{-sA} - e^{sA} B A e^{-sA} = e^{sA} [A,B] e^{-sA}
f''(s) = e^{sA} A [A,B] e^{-sA} - e^{sA} [A,B] A e^{-sA} = e^{sA} [A, [A,B]] e^{-sA}
f'''(s) = e^{sA} [A, [A, [A,B]]] e^{-sA}
etc.
Now construct the Taylor series for f(s):
f(s) = f(0) + s f'(0) + \frac12 s^2 f''(0) + \frac1{3!} s^3 f'''(0) + ...
e^{sA} B e^{-sA} = B + [A,B] s + \frac12 [A, [A, B]] s^2 + \frac1{3!} [A, [A, [A, B]]] s^3 + ...
Finally, evaluate the above at s=1 to get the result.
