Homogeneous Structures on $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$

We determine all the homogeneous structure tensors on $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$. This work together with previous articles yields a complete classification of all the homogeneous structure tensors on three-dimensional homogeneous Riemannian manifolds.