On Euler's magic matrices of sizes $3$ and $8$

A proper Euler’s magic matrix is an integer $n\times n$ matrix $M\in {\mathbb{Z}}^{n\times n}$ such that $M\cdot M^t=\gamma \cdot I$ for some nonzero constant $\gamma$, the sum of the squares of the entries along each of the two main diagonals equals $\gamma$, and the squares of all entries in $M$ are pairwise distinct. Euler constructed such matrices for $n=4$. In this work, we construct examples for $n=8$ and prove that no such matrix exists for $n=3$.