The Lorentz group is a fundamental group in theoretical physics that describes the symmetries of spacetime in special relativity. Named after the Dutch physicist Hendrik Lorentz, it consists of all linear transformations that preserve the spacetime interval between events in Minkowski space. In mathematical terms, the Lorentz group can be defined as the set of all Lorentz transformations, which are transformations that can be expressed as linear transformations of the coordinates in spacetime that preserve the Minkowski metric.
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Subgroup of the Poincaré group without translations. Therefore, in those, the spacetime origin is always fixed.
Or in other words, it is as if two observers had their space and time origins at the exact same place. However, their space axes may be rotated, and one may be at a relative speed to the other to create a Lorentz boost. Note however that if they are at relative speeds to one another, then their axes will immediately stop being at the same location in the next moment of time, so things are only valid infinitesimally in that case.
This group is made up of matrix multiplication alone, no need to add the offset vector: space rotations and Lorentz boost only spin around and bend things around the origin.
One definition: set of all 4x4 matrices that keep the Minkowski inner product, mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) page 63. This then implies: