I'm trying to answer the following question:
Is any positive and constant metric in $\Re³$ equivalent to the usal metric defined as $$ds² = dx² + dy² + dz² \tag{*}\label{1} $$ with $ds = g_{ij}dx^{i}dx^{j}$ and $g_{ij} = \delta_{ij}$. Or alternatively, does always exist a coordinate transformation from the coordinates $y^\mu$ to the coordinates $x^i$ such that \eqref{1} is recovered from $$ds² = \tilde{g}_{\mu\nu}dy^{\mu}dy^{\nu} $$ with $\tilde{g}_{\mu\nu}$ being constant in the $y^\mu$ coordinates.
My initial guess it's that this is true, however I'm not sure on how this could be proved.
Any symmetric matrix can be diagonalized by an orthogonal matrix, say $O$ - so $O^{T}gO = \Lambda$, $\Lambda$ diagonal with constant eigenvalues. We have positive eigenvalues, and since $O$ is constant, $dy^i = O^i_{j}dx^j$. Now our metric is diagonal - we scale each variable by the square root of its corresponding eigenvalue; that will turn the metric to the identity.
Note this doesn't work if the metric has negative eigenvalues.