A proper Euler magic matrix of order $5$

An Euler magic matrix is an integer matrix $M$ with $MM^{t}=\gamma I$ whose squared entries sum to $\gamma$ along both main diagonals; it is proper if its squared entries are pairwise distinct.

Euler constructed an order-$4$ proper example, and Müller settled orders $3$ (none exist) and $8$, leaving order $5$ as the smallest open case.

We construct such a matrix, by rotating one of Müller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.