= Solution
Write $n=bk$, let $I_n$ be the <identity matrix>, let $J_m$ be the $m$-by-$m$ <all-ones matrix>, and put $B=\operatorname{diag}(J_k,\ldots,J_k)$, with one diagonal block per experimental block. The <variance-covariance matrix> is
$$
\Sigma=\sigma^2\{(1-\rho_1)I_n+(\rho_1-\rho_2)B+\rho_2J_n\}.
$$
Its <orthogonal decomposition> is obtained by first subtracting each block's <sample mean>, then subtracting the grand <sample mean> from the block means. More explicitly, writing $x_{jt}$ for coordinate $t$ in block $j$, the three invariant subspaces are
$$
W=\{x:\sum_{t=1}^kx_{jt}=0\text{ for every }j\},\qquad
U=\{x:x_{jt}=a_j,\ \sum_{j=1}^ba_j=0\},\qquad
G=\operatorname{span}\{\mathbf1_n\}.
$$
They are pairwise <orthogonal>, with <dimensions> $b(k-1)$, $b-1$ and $1$, and their <direct sum> is $\mathbb R^n$. On $W$, both $B$ and $J_n$ vanish. On $U$, $B=kI$ and $J_n=0$. On $G$, $B=kI$ and $J_n=nI$. Thus the \b[<eigenvalues> and corresponding invariant subspaces] are
$$
\boxed{\begin{array}{c|c|c}
\text{subspace}&\text{eigenvalue}&\text{dimension}\\
W&\sigma^2(1-\rho_1)&b(k-1)\\
U&\sigma^2[1+(k-1)\rho_1-k\rho_2]&b-1\\
G&\sigma^2[1+(k-1)\rho_1+k(b-1)\rho_2]&1
\end{array}}
$$
If some <eigenvalues> coincide, their <eigenspace> is the <direct sum> of the listed subspaces with that value. Zero-dimensional rows are omitted when $b=1$ or $k=1$. For an admissible <covariance matrix> the <eigenvalues> on nonzero subspaces must be nonnegative; these conditions are also sufficient for <positive semidefiniteness>. The <expectation> parameters do not affect this calculation.
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