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.