Assuming I have the $N$-dimensional irreducible representation of $SU(2)$ (The $N$-dimensional spin matrices $Sx$,$Sy$ and $Sz$) and I can consruct arbitrary polynomials from them of the form:
$$\sum{c_{nmk}S_x^nS_y^mS_z^k}$$ (allowing for alternating orders as well) Can I generate any arbitrary unitary matrix in $N$ dimensions? Meaning, does any arbitrary unitary can be written as $$U=e^{i\sum{c_{nmk}S_x^nS_y^mS_z^k}}?$$ I think the question can be alternatively written as, can any arbitrary nxn hermitian matrix be written as that polynomial.
Is there any standart proof?
I apologize for using notations from physics in case they are unclear.