Some sources distinguish "invariant" from "covariant" such that under some transformation (typically Lie group):TODO examples.

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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:

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TODO, including why the Schrodinger equation is not.

Physics from Symmetry by Jakob Schwichtenberg (2015) page 66 shows one in terms of 4x4 complex matrices.

More importantly though, are the representations of the Lie algebra of the Lorentz group, which are generally also just also called "Representation of the Lorentz group" since you can reach the representation from the algebra via the exponential map.

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One of the representations of the Lorentz group that show up in the Representation theory of the Lorentz group.

Predicts fine structure.

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TODO understand a bit more intuitively.

Two observers travel at fixed speed relative to each other. They synchronize origins at x=0 and t=0, and their spacial axes are perfectly aligned. This is a subset of the Lorentz group. TODO confirm it does not form a subgroup however.

Generalization of orthogonal group to preserve different bilinear forms. Important because the Lorentz group is $SO(1,3)$.

Like the special orthogonal group is to the orthogonal group, $SO(m,n)$ is the subset of $O(m,n)$ with determinant equal to exactly 1.

More interestingly, how is that implied by the Stern-Gerlach experiment?

physics.stackexchange.com/questions/266359/when-we-say-electron-spin-is-1-2-what-exactly-does-it-mean-1-2-of-what/266371#266371 suggests that half could either mean:

A bit like the classification of simple finite groups, they also have a few sporadic groups! Not as spectacular since as usual continuous problems are simpler than discrete ones, but still, not bad.

Recommended from Physics from Symmetry by Jakob Schwichtenberg (2015) page 92:

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