24

Now let us take the group:

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The coadjoint action is:

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c
'

l m c

Same calculation scheme.
But this time, we will recover a product

l m

Once again, when you differentiate
the matrix

g

, do not differentiate either

l

or

m

. The orthochronous Lorentz group Lo has two components. Introducing

l

= ± 1 and

m

= ± 1 increases the number of components to:

2 × 2 × 2 = 8

This group now contains retrochronous components.

The following diagrams
show the motions and the coadjoint action, with the region
where the element

g

was chosen indicated in gray.

First, the "playing field":

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A certain number of symmetries can be defined
from this graph.

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This grayed region corresponds to the orthochronous subgroup of the extended Poincaré group.
Below, in the sectors, a motion of a matter particle is shown. These subgroup elements lead to
other motions, which also correspond to matter.

These elements can
also act on the motion of a photon. See Figure 1 bis.

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