orthogonal group SO(n) (generalizing the rotation group SO(3) for ). It follows that the n-dimensional hyperbolic space can be exhibited as the homogeneous space and a Riemannian symmetric space of rank 1, The group SO+(1,n) is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space. In more concrete terms, SO+(1,n) can be split into n(n-1)/2 rotations (formed with a regular Euclidean rotation matrix in the lower-right block) and n hyperbolic translations, which take the form where is the distance translated (along the x axis in this case), and the 2nd row/column can be exchanged with a different pair to change to a translation along a different axis. The general form of a translation in 3 dimensions along the vector is: where . This extends naturally to more dimensions, and is also the simplified version of a Lorentz boost when you remove the relativity-specific terms. Examples of groups of isometries The group of all isometries of the hyperboloid model is O+(1,n). Any group of isometries is a subgroup of it. Reflections For two points , there is a unique reflection exchanging them. Let . Note that , and therefore . Then is a reflection that exchanges and . This is equivalent to the following matrix: (note the use of block matrix notation). Then is a group of isometries. All such subgroups are conjugate. Rotations and reflections is the group of rotations and reflections that preserve . The function is an isomorphism from O(n) to this group. For any point , if is an isometry that maps to , then is the group of rotations and reflections that preserve . Translations For any real number , there is a translation This is a translation of distance in the positive x direction if or of distance in the negative x direction if . Any translation of distance is conjugate to and . The set is the group of translations through the x-axis, and a group of isometries is conjugate to it if and only if it is a group of isometries through a line. For example, let's say we want to find the group of translations through a line . Let be an isometry that maps to and let be an isometry that fixes and maps to . An example of such a is a reflection exchanging and (assuming they are different), because they are both the same distance from .