= Solution
{c}
<Independence> of the centered <standard normal random variables> gives $\mathbb E\epsilon_i=0$ for every $i$. Also
$$
\mathbb E\epsilon_1^2=1,\qquad
\mathbb E\epsilon_i^2=\mathbb E\eta_i^2\,\mathbb E\eta_{i-1}^2=1\quad(i\ge2).
$$
For nonadjacent indices, the underlying sets of $\eta$ variables are disjoint, so the errors are <independent> and have zero <covariance>. For adjacent indices with $i\ge2$,
$$
\mathbb E[\epsilon_i\epsilon_{i+1}]
=\mathbb E[\eta_{i-1}\eta_i^2\eta_{i+1}]
=\mathbb E\eta_{i-1}\,\mathbb E\eta_i^2\,\mathbb E\eta_{i+1}=0.
$$
The first adjacent pair separately gives $\mathbb E[\epsilon_1\epsilon_2]=\mathbb E[\eta_1^2\eta_2]=0$. Hence
$$
\boxed{\Sigma=I_n.}
$$
These <adjacent products of independent Gaussian variables> are <uncorrelated random variables> even though adjacent errors share a factor and are dependent.
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