To prove

The transformation is a Klein four-group

Proof:

We provide evidence using a random example of three items (a triad) who are related by a transformation:

• a ↔ ab is not different from the bracket expression <b>

• a ↔ a<b> is not different from the bracket expression ab

• ab ↔ a<b> is not different from the bracket expression a

This is easily checked by writing the transformation as a well-formed bracket expression and to reduce the latter. Indeed, a transformation of two points always determines a unique third point.

We notice now that each of these points is not different from their transformation with <>, where we move from the potential lattice to collapsed lattice.

With these data we can construct a Cayley table for the four points involved in the transformation and each row and each column shows these points only once.

This table is the evidence that the triad with <> forms a closed structure: the Klein four-group. The four group of Klein in fact is defined as the structure with I² = J² = (IJ)² = 1 (or alternatively with (IJ) = K: I² = J² = K² = 1), and with a classical example: for I the rotation by π around the spatial x-axis, for J the rotation by π around the spatial y-axis, for K the rotation by π around the spatial z-axis.

QED

A fully equivalent table as the table above is given by the embedding of all points, which shows once more the fundamental duality in the bracket formalism.

The Cayley table gives an overview of the structures that one might be able to assign to the value of <> or to the value <<>> by a choice of two well-formed bracket expressions between which one transforms.

In the two distinctions universe one finds six four-groups in a and <>:

Note that with one distinction also one Klein four-group is generated