A Lorentz boost along I acts on a Dirac-level four-vector a = a0β0 + a·β by one similarity, aL = ΛaΛ−1, with Λ = [[U−1, 0], [0, U]] built from the Pauli rotor U = eψσI/2. Choose a four-vector and a velocity; the justification from the Pauli boost and the lifted Pauli results follow below.
Take the BQ-Pauli boost as known: AL = U−1AU−1, and for the time adjoint (AL)T = UATU. The Dirac vector carries A and −AT as its two blocks, so its boost follows in one line:
The two Pauli rules, neither of them a similarity, are the two blocks of one similarity. Λ is fixed by the upper block (up to a common factor, which cancels), and the lower block then comes out as the Pauli adjoint rule by itself.
The boost slides the point along its hyperbola: a0 ± a1 are multiplied by e∓ψ, the Doppler factors, and a2, a3 are unchanged. The invariant is the square aa = −(a0² − a²) 𝟙.
Written as a rotor, Λ = e−ψαI/2, the boost generalizes by replacing αI: e−ψn·α/2 boosts along any n, e−θm·Σ/2 rotates about m, and two boosts in different directions compose into a boost and a Thomas–Wigner rotation. These go beyond the papers; the velocity is the one set above.
As at the Pauli level, the diagonal Λ belongs to a boost along I: the frames are aligned so that their relative velocity lies along I (Synge’s communal photon), and the Doppler factors e±ψ/2 sit on single diagonal entries. A boost in another direction can be replaced by a rotation and an I-boost, which is where the Thomas–Wigner rotation enters; the last section shows the same in rotor form. All of this is in the Weyl representation; for the Dirac representation see Lorentz transformations: Weyl and Dirac.
The Pauli boost, the rapidity, (AL)T = UATU and Lemma 2.1 follow the BQ-Pauli papers; Λ = diag(U−1, U) and XL = ΛXΛ−1 follow de Haas (2026), BQ-Weyl electromagnetism, eqs. 161–196. Eq. 161 writes U = exp((ψ/2)Î), which is the phase diag(eiψ/2, e−iψ/2); the matrix Λ of eq. 163 needs U = exp((ψ/2)σI), used here. Λνμ (with indices) is the real coordinate matrix, Λ (without) the 4×4 rotor.
Every identity is checked exactly in bq_dirac_lorentz.py. Part of the BQ algebra at the Dirac level; the Pauli version is BQ Lorentz transformations. The same boost in the Dirac representation, ΛD = SΛWS−1, is on Lorentz transformations: Weyl and Dirac.