= Solution
Treat the site <atomic orbitals> $s_1,\ldots,s_4$ as orthonormal, as in the <Hückel method>. A normalized set of <symmetry-adapted linear combinations> is
$$
\boxed{\begin{aligned}
\Phi_{A_{1g}}&=\tfrac12(s_1+s_2+s_3+s_4),\\
\Phi_{B_{1g}}&=\tfrac12(s_1-s_2+s_3-s_4),\\
\Phi_{E_u,x}&=\tfrac1{\sqrt2}(s_3-s_1),\\
\Phi_{E_u,y}&=\tfrac1{\sqrt2}(s_2-s_4).
\end{aligned}}
$$
The first combination is unchanged by every operation; the alternating combination has the transformation signs of $x^2-y^2$ for the printed axes. The last two transform together like $(x,y)$. The squared coefficients sum to one in each case, and their pairwise inner products vanish. Any orthogonal rotation of the two $E_u$ combinations is an equally valid <basis> for that degenerate subspace.
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b-square-orbitals.png]
{title=Original coefficient and phase maps for the four square-array symmetry orbitals}
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