= Solution
Parameters $\psi$ and $\lambda$ are <orthogonal parameters> if the cross block of the expected <Fisher information matrix> is zero throughout the relevant parameter region:
$$
I_{\psi\lambda}=\mathbb E_\theta[\ell_\psi\ell_\lambda^T]=-\mathbb E_\theta\ell_{\psi\lambda}=0.
$$
This is a statement about expected information, not a requirement that every observed cross derivative vanish. For regular <maximum-likelihood estimators>, $\sqrt n(\widehat\theta-\theta)$ has limiting <covariance> equal to the inverse per-observation <Fisher information matrix>. Block diagonality therefore gives asymptotically <independent> fitted interest and <nuisance parameters>. The efficient information $I_{\psi\psi}-I_{\psi\lambda}I_{\lambda\lambda}^{-1}I_{\lambda\psi}$ reduces to $I_{\psi\psi}$ at an orthogonal parametrization, although nuisance estimation can still matter at higher orders.
For scalar interest and nuisance, retain $\psi$ and write the old nuisance as $\lambda=\lambda(\psi,\eta)$. The transformed cross information is $\lambda_\eta(I_{\psi\lambda}+\lambda_\psi I_{\lambda\lambda})$. Hence an orthogonal nuisance coordinate can locally be found from
$$
\boxed{\lambda_\psi=-I_{\psi\lambda}/I_{\lambda\lambda},\qquad I_{\lambda\lambda}>0,}
$$
subject to local solvability and a nonsingular coordinate change. Global coordinates need not follow from this local construction. For the <normal distribution> with separately varying mean and <variance> parameters $(\mu,v)$, the scores are $(Y-\mu)/v$ and $-1/(2v)+(Y-\mu)^2/(2v^2)$; their product has zero <expectation> by symmetry. Thus the per-observation information is diagonal with entries $1/v$ and $1/(2v^2)$.
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