In the main article, the new Cartesian basis \(\{\ep, i\ep, \em, i\em\}\) was considered as a basis of a commutative algebra constructed from the idempotents \(\ep\) and \(\em\). This form is particularly useful for geometric interpretation because it explicitly separates two independent complex planes of the internal phase space.

This appendix considers another, equivalent form of the same commutative structure: the algebra \(\mathcal{A}\) with basis \(\{1,i,j,ij\}\), where \(i^2=-1\), \(j^2=1\), and \(ij=ji\). It expresses the same elements in terms of the identity, the imaginary unit, the hyperbolic unit, and their product, rather than in terms of idempotent components. The purpose of the appendix is therefore not to replace a noncommutative algebra with a commutative one, but to compare two commutative representations and derive explicit transformations between them.

The first representation is idempotent. It uses the basis:

Here, \(\ep\) and \(\em\) are idempotents, and \(i\) is the usual imaginary unit. The basic multiplication rules are:

These rules immediately imply that the two components associated with \(\ep\) and \(\em\) do not mix. An arbitrary element can therefore be written as the sum of two independent complex parts:

The second representation uses the algebraic basis:

Multiplication in this basis is defined by the rules:

Consequently, this is also a commutative associative algebra over \(\mathbb{R}\). An arbitrary element has the form:

Both expressions describe the same structure, since the algebra is isomorphic to the direct sum of two copies of the complex numbers:

This is the key property: the algebra splits into two independent complex components, while the different bases simply represent this decomposition in different ways. The algebraic condition \(\ep\em=0\) separates the channels; metric orthogonality requires an additional choice of inner product.

The two forms are related through the hyperbolic unit \(j\), for which \(j^2=1\). The idempotents are expressed in terms of it as follows:

Conversely, the hyperbolic unit is expressed in terms of the idempotents:

For the product \(ij\), we obtain:

For the complex idempotent directions, we have:

Thus, the basis \(\{1,i,j,ij\}\) and the basis \(\{\ep,i\ep,\em,i\em\}\) can be expressed as linear combinations of one another.

The same element \(Z\) can be written in the new Cartesian basis using four real coefficients \(\xi_0,\xi_1,\xi_2,\xi_3\). These denote internal phase coordinates, rather than laboratory spacetime coordinates. Their physical dimension is determined by the chosen object \(Z\); all four coefficients have the same dimension:

or in the algebraic basis:

Substituting (8) and (11) into representation (12), we obtain:

The forward transformation from the coordinates \((\xi_0,\xi_1,\xi_2,\xi_3)\) to the coefficients \((A,B,C,D)\) is therefore:

The inverse transformation is:

In matrix form, the forward transformation is written as:

The inverse transformation has the form:

Equations (8)–(11) imply that the element \(Z=A+Bi+Cj+D(ij)\) can be represented as a pair of complex numbers:

The first complex component corresponds to the idempotent \(\ep\), and the second to the idempotent \(\em\):

In terms of the coordinates \((\xi_0,\xi_1,\xi_2,\xi_3)\), this decomposition takes a particularly simple form:

Equation (21) illustrates the main advantage of the idempotent basis: an element of the internal phase space immediately splits into two complex components represented in the planes \((\xi_0,\xi_1)\) and \((\xi_2,\xi_3)\). The basis \(\{1,i,j,ij\}\) expresses the same structure in a more symmetric algebraic form.

In the main article, the operator \(J(a,b)\) combines the internal phase with the external motion parameter. Consider the special case in which the center is at rest: \(b=0\) and \(a=\omega t/\pi\), where \(t\) is time in the rest frame of the center, and \(\omega>0\) is the constant angular frequency of the internal component with respect to that time. The quantity \(V_J=cJ\) introduced in the model describes phase-geometric dynamics; it is not a laboratory velocity vector:

where \(e^{i\omega t}=\cos\omega t+i\sin\omega t\). In the idempotent basis, this gives:

Changing to the basis \(\{1,i,j,ij\}\), we obtain:

The same equation can be written using the half-angle identities:

Define the internal integrated curve \(R_J(t)=\int_0^t V_J(t')\,dt'\), so that \(R_J(0)=0\). In equation (12), we now set \(Z=R_J(t)\); its coefficients \(\xi_k(t)\) have the dimension of length. Integration gives:

In the coordinate representation of the internal space, equation (26) defines a helical curve: the coefficient \(\xi_0\) increases linearly, while the pair \((\xi_2,\xi_3)\) describes a circle of radius \(c/\omega\) centered at \((0,c/\omega)\). The coefficient \(\xi_0=ct\) results from integrating the constant phase component; it is not identified with the laboratory time coordinate. The curve \(R_J\) is not the worldline of the particle center. The fact that its coefficients have the dimension of length does not by itself establish a physical distance. Both forms of the algebra yield the same internal curve; its physical interpretation requires a separate mapping.

The algebraic normalization of the operator must also be distinguished from the scalar norm of a state. For \(J=\ep+\em e^{i\omega t}\), we have \(J\overline J=1\), where conjugation reverses the sign of \(i\) and leaves the idempotents unchanged. If the state \(\Psi=\ep u+\em v\) is assigned the Hilbert-space norm \(\|\Psi\|^2=|u|^2+|v|^2\), multiplication by \(J\) preserves that norm. However, \(J\) itself, considered as a vector with coefficients \((1,e^{i\omega t})\), has squared norm \(2\) in this metric. The identity \(J\overline J=1\) must therefore not be confused with a unit Hilbert-space norm; nor does it by itself constitute a law of conservation of physical energy.

The observed motion of the center is determined by the external mapping specified in the main article:

Here, \(s\) is an oriented coordinate along the physical trajectory, and \(\widehat{\mathbf t}(s)\) is its unit tangent vector. In the example considered, \(b=0\), so \(\mathbf v=0\) and the center remains at rest, although the internal phase and the curve \(R_J\) continue to evolve. Transformations (15)–(18) are changes of basis within the same algebra; by themselves, they define neither Lorentz transformations nor the metric of physical spacetime.

1. Clear geometry. The basis \(\{\ep,i\ep,\em,i\em\}\) explicitly shows the decomposition of the internal phase space into two complex planes: \((\xi_0,\xi_1)\) and \((\xi_2,\xi_3)\).

2. Natural representation of internal dynamics. The quantity \(V_J(t)=c(\ep+\em e^{i\omega t})\) has a compact form: the first component is constant, while the second contains a periodic internal phase. After integration, they produce the linear and circular parts of the internal curve.

3. Diagonal separation of components. Since \(\ep\em=0\), the two idempotent parts do not mix under multiplication. This makes transformations in the independent planes particularly transparent.

4. Explicit internal coordinates. In this basis, the coordinates \(\xi_0,\xi_1,\xi_2,\xi_3\) appear directly as coefficients of the corresponding basis elements, making the internal coordinate representation easy to visualize. The observed coordinates of the center are determined by a separate physical mapping.

1. Unified algebraic notation. The basis \(\{1,i,j,ij\}\) uses familiar units: the real identity \(1\), the imaginary unit \(i\), the hyperbolic unit \(j\), and their product \(ij\).

2. Explicit connection between complex and hyperbolic structures. This form clearly shows that two types of units coexist within a single commutative algebra: \(i^2=-1\) and \(j^2=1\).

3. Convenient treatment of general transformations. The representation in terms of \(A+Bi+Cj+Dij\) is useful for deriving change-of-basis matrices, comparing different algebras, and analyzing symmetries.

4. Commutativity is preserved. Unlike quaternionic models, the order of the factors does not matter here: \(ij=ji\). This simplifies calculations involving phase factors and powers.

5. Equivalence to \(\mathbb{C}\oplus\mathbb{C}\). The representation \(\{1,i,j,ij\}\) explicitly shows that the entire construction can be viewed as two independent complex components. The isomorphism itself does not define any physical interaction between the channels.

This appendix has shown that the new Cartesian basis \(\{\ep,i\ep,\em,i\em\}\) and the algebraic basis \(\{1,i,j,ij\}\) are two equivalent commutative representations of the same structure. Both bases yield the decomposition \(\mathcal{A}\cong\mathbb{C}\oplus\mathbb{C}\), while emphasizing different aspects of the algebra.

The idempotent representation is useful for the geometric interpretation of the internal state because it directly separates the phase space into the planes \((\xi_0,\xi_1)\) and \((\xi_2,\xi_3)\). The representation \(\{1,i,j,ij\}\) emphasizes the common algebraic structure, the connection between complex and hyperbolic units, and compact expressions for coordinate transformations.

Thus, the appendix does not replace one algebra with another; it demonstrates the advantages of both notations. The idempotent basis is useful for visualizing the geometry of the internal state, while the basis \(\{1,i,j,ij\}\) is useful for working with the general algebraic structure and transformations between representations. A change of basis preserves the mathematical object, but does not replace the mapping of that object into observable kinematics. This distinction brings the appendix into agreement with the phase-space interpretation adopted in the main article.