The Lorentz group is the group of time-preserving linear isometries of Minkowski space with the Minkowski metric

(where the convention is used). It is also the group of isometries of three-dimensional hyperbolic geometry. It is time-preserving in the sense that the unit time vector is sent to another vector such that .

A consequence of the definition of the Lorentz group is that the full group of time-preserving isometries of Minkowski space is the group direct product of the group of translations of (i.e., itself, with addition as the group operation), with the Lorentz group, and that the full isometry group of the Minkowski is a group extension of by the product .

The Lorentz group is invariant under space rotations and Lorentz transformations.