I'm aware of Bianchi's classification of homogenous 3-manifolds into the Bianchi types I through IX, and I can follow the algebra for classifying the Lie algebras. However, I still can't visualize the different spaces, except for the simpler ones. For example, I can see that type I is just locally Euclidean $\mathbb{E}^3$. Other homogenous spaces that I can visualize are locally spherical $S^3$, which I think corresponds to type IX? I can further imagine $S^2\times\mathbb{R}$ as a homogenous space, but I can't tell if that corresponds to one or more Bianchi types by combining the two spaces in different ways.

How do I go about understanding the geometry of the different Bianchi types and visualizing them?

I'm aware of Bianchi's (local) classification of homogenous 3-manifolds into the Bianchi types I through IX, and I can follow the algebra for classifying the Lie algebras. However, I still can't visualize the different spaces, except for the simpler ones. For example, I can see that type I is just locally Euclidean $E^3$. Other homogenous spaces that I can visualize are locally spherical $S^3$, which I think corresponds to type IX? I can further imagine $S^2\times\mathbb{R}$ as a homogenous space, but I can't tell if that corresponds to one or more Bianchi types by combining the two spaces in different ways.

How do I go about understanding the (local) geometry of the different Bianchi types and visualizing them?

EDIT: Added (local) above. I am not asking anything topological, I am just interested in the local geometry.