In brief. In \(≤4\) dimensions, there’s a simple formula for the bivector \(σ_3\) in terms of bivectors \(σ_1\) and \(σ_2\) such that \(e^{σ_1}e^{σ_2} = ±e^{σ_3}\).

Abstract. We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations \(e^{σ_i}\) in the spin representation (a.k.a. Lorentz rotors) in terms of their generators \(σ_i\):

\[\ln(e^{σ_1}e^{σ_2}) = \tanh^{-1}\qty(\frac{ \tanh σ_1 + \tanh σ_2 + \frac12\qty[\tanh σ_1, \tanh σ_2] }{ 1 + \frac12\qty{\tanh σ_1, \tanh σ_2} })\]

This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension \(≤ 4\), naturally generalising Rodrigues’ formula for rotations in \(ℝ^3\). In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex \(2×2\) matrix representation realised by the Pauli spin matrices. The formula is applied to the composition of relativistic \(3\)-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

Full paper. At doi.org/10.1142/S0219887821502261 and on Arχiv.