I'm trying to understand how one can obtain $\mathbb{M}^4$ as the quotient space

$$

ISO(3,1)/SO(3,1),

$$

and equivalently de Sitter and Anti-de Sitter as $SO(4,1)/SO(3,1)$ and $SO(3,2)/SO(3,1)$, respectively. I.e. we can see these manifolds by taking their symmetry Lie group and quotienting by the normal Lorentz subgroup $SO(3,1)$.

This construction seems intuitive but I cannot find a decent reference where these steps are taken with a little more care. I would like to see $\mathbb{M}^4$ emerging from explicit group theoretical calculations (I'm not so familiar with group actions, orbits and all that).

Can anyone provide a nice and specific explanation without referencing to general theorems but starting from general grounds?

I'm not interested in deepening the construction coming from topology.. I know it is a general fact that one can construct a manifold given a transitive action of a Lie group, as the quotient of the Lie group by the stabilizer of a point.

Thank you!