= Solution
Each entry of the <matrix> $Q$ from part (b)(iii) has unique real and imaginary <integer> coefficients. Write
$$
Q=U+iV,\qquad U,V\in M_2(\mathbb Z).
$$
It already belongs to the <general linear group> $\operatorname{GL}_2(\mathbb Z[i])$. Expanding the relation from part (b)(iii) gives
$$
0=(iI-A)(U+iV)e_2=(-AU-V)e_2+i(U-AV)e_2.
$$
The real and imaginary parts must both vanish, so
$$
\boxed{AUe_2=-Ve_2,\qquad AVe_2=Ue_2,\qquad U+iV\in\operatorname{GL}_2(\mathbb Z[i])}.
$$
These are the required identities, with $e_2=(0,1)^{\mathsf T}$.
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