Consider the bilinear form $q(x,y) = - x_{n+1} y_{n+1} + \sum_{j=1}^{n} x_{j}y_{j}$ for $x, y \in \mathbb{R}^{n+1}$.
Consider the set $T = \{ x \in \mathbb{R}^{n+1} | q(x,x) = - 1, x_{n+1} > 0 \}$ (this is the set of timelike vectors, in the physics sense).
Can I give $T$ a group structure?
I feel like I should be able to, because $T \approx SO^{+}(n,1) / SO(n)$, but I can't figure out what the group operation would be.
I think what you are asking about is a way to perform addition on velocities in special relativity with $n$ spatial coordinates. From what I gather from a Wikipedia article on the subject, there exists a, supposedly natural, way to perform such an addition, but at least for $n=3$ it is neither commutative nor associative, so in particular this does not give a group structure. It would seem that the phenomenon of Thomas precession reflects that fact that one cannot separate out a subgroup of "pure translations" in the Lorentz group. Which mathematically no doubt is related to $SO(n)$ not being normal in $SO^+(n,1)$.