Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures

We classify the hypersurfaces of $\mathbb{Q}^3\times\mathbb{R}$ with three distinct constant principal curvatures, where $\varepsilon \in \{1,-1\}$ and $\mathbb{Q}^3$ denotes the unit sphere $\mathbb{S}^3$ if $\varepsilon = 1$, whereas it denotes the hyperbolic space $\mathbb{H}^3$ if $\varepsilon = -1$. We show that they are cylinders over isoparametric surfaces in $\mathbb{Q}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of $\mathbb{Q}^3\times\mathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in $\mathbb{Q}^3\times\mathbb{R}$.