  In this paper, firstly, we show the existence of a compact embedded constant
mean curvature (CMC) hypersurface $\Sigma_1$ in $\mathbb{S}^{2n}$ of the type
$S^{n-1} \times S^{n-1} \times S^{1}$. Moreover, the hypersurface $\Sigma_1$
exhibits $O(n)\times O(n)$ symmetry. Secondly, we show that there exists a
compact embedded CMC-hypersurface $\Sigma_2 \subset \mathbb{S}^{3n-1}$ of the
type $S^{n-1} \times S^{n-1} \times S^{n-1} \times S^{1}$. These results
generalize the results of Carlotto and Schulz.
