= Solution
\b[False.] Consider the real <matrix>
$$
A_1=\begin{pmatrix}-1&1&0\\0&c&-c\\0&c&c\end{pmatrix},\qquad c=\frac1{\sqrt2}.
$$
It is block upper triangular, so its <characteristic polynomial> is the same as that of $A_0$ and it has the required <eigenvalues>. Its second column, however, has squared <Euclidean norm> $1+c^2+c^2=2$, whereas every column of an <orthogonal matrix> has norm one. Thus
$$
\boxed{A_1^{\mathsf T}A_1\ne I}.
$$
Unit-modulus <eigenvalues> alone do not imply orthogonality; they do not control the geometry of the <eigenvectors>.
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