= Solution
For any two distinct normalized <Majorana fermion operators> $\eta,\chi$, let $f=(\eta+i\chi)/2$. Then
$$
\{f,f^\dagger\}=\frac14\big(\{\eta,\eta\}+\{\chi,\chi\}\big)=1,
$$
and
$$
\{f,f\}=\frac14\big(\{\eta,\eta\}-\{\chi,\chi\}+2i\{\eta,\chi\}\big)=0.
$$
The same applies to $f^\dagger$. Both specified pairs have this form. Moreover, the Majorana operators entering $f_1$ are all distinct from those entering $f_2$, so every cross <anticommutator> is zero. Therefore
$$
\boxed{\{f_i,f_j^\dagger\}=\delta_{ij},\qquad
\{f_i,f_j\}=\{f_i^\dagger,f_j^\dagger\}=0}.
$$
Thus $f_1,f_2$ are ordinary <fermionic annihilation operators>, constructed by a different pairing of the four Majorana operators.
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