= Solution
There is a normalization mismatch in the printed definitions: $\sigma^x\pm i\sigma^y$ have nonzero entries of magnitude two, whereas a single fermionic creation or annihilation operator has magnitude one. To obtain the intended <Jordan–Wigner transformation>, use normalized spin ladders $s_j^\pm=(\sigma_j^x\pm i\sigma_j^y)/2$. The given fermion expressions represent these $s_j^\pm$. The <two-site Jordan-Wigner normalization and signs> then fix every factor unambiguously.
Put $n_j=f_j^\dagger f_j$ and $P_1=e^{i\pi n_1}=1-2n_1$. The <canonical anticommutation relations> give $P_1^2=I$, $P_1f_1=-f_1P_1$, and $[P_1,f_2]=[P_1,f_2^\dagger]=0$. Hence
$$
[s_1^-,s_1^+]=[f_1,f_1^\dagger]=1-2n_1=-\sigma_1^z,\qquad
[\sigma_1^z,\sigma_2^z]=4[n_1,n_2]=0,
$$
$$
[s_1^-,s_2^-]=f_1P_1f_2-P_1f_2f_1=-P_1\{f_1,f_2\}=0,
$$
and
$$
[s_2^-,s_2^+]=[P_1f_2,f_2^\dagger P_1]=f_2f_2^\dagger-f_2^\dagger f_2=1-2n_2=-\sigma_2^z.
$$
These are the <spin commutation relations>, with distinct sites commuting. If one retains the unscaled symbols $\sigma_j^\pm=2s_j^\pm$, the first and last displayed commutators instead equal $-4\sigma_j^z$; the off-site commutators remain zero. Setting the unscaled $\sigma_j^-$ equal to $f_j$ would contradict the original <Pauli matrices>.
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