Let $R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:

(I) Trace $(R)=0$

(II) If trace $(\operatorname{adj}(\operatorname{adj}(R))=0$, then $R$ has exactly one non-zero entry.