= Solution
Let $Z_{ir}=\mathbf1_{\{\text{observation }i\text{ belongs to rat }r\}}$. After integrating out the <random intercepts>,
$$
Y\sim N_{176}(X\beta,V),\qquad V=\tau^2ZZ^T+\sigma^2I_{176}.
$$
The fixed-effect <design matrix> has four independent columns, so the given $A$ has dimensions $176\times172$, with $A^TA=I_{172}$ and $A^TX=0$. Consequently the error contrasts $U=A^TY$ have a <multivariate normal distribution> with mean zero and <covariance matrix> $W=A^TVA$.
The <restricted likelihood from orthogonal error contrasts> is their density. Equivalently, the estimates maximize
$$
\boxed{\ell_R(\tau^2,\sigma^2)=-\frac{172}{2}\log(2\pi)-\frac12\log\det W-\frac12Y^TAW^{-1}A^TY,\qquad\tau^2\geq0,\quad\sigma^2>0.}
$$
This is <restricted maximum likelihood>, not maximization over individual rat effects. The unknown $\beta$ disappears because $A^TX=0$. Replacing $A$ by $AQ$ for an orthogonal $Q$ rotates $U$ and conjugates $W$, preserving the <determinant> and quadratic form; hence the objective is independent of the chosen <orthonormal basis>. The software's integrated REML log-likelihood can differ by an additive constant depending only on $X$, which has no effect on these variance estimates.
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