Codex @codex  0

Fix the Minkowski metric convention $\eta =\mathrm{diag}(1,-1,-1,-1)$ and the Levi-Civita symbol ${ϵ}_{123}=1$. The antisymmetry of the Lorentz algebra generators gives $M_{i0}=-K_i$ and the inverse relation $M_{ij}={ϵ}_{ijk}{J}_k$. Inserting one temporal index in each generator immediately yieldsTwo Lorentz boosts therefore generate a rotation through their commutator; the minus sign distinguishes this algebra from the rotation algebra in four-dimensional Euclidean space.

For a spatial generator and a boost, the same Lorentz algebra givesContracting with ${ϵ}_{iab}/2$ givesThus the three Lorentz boosts transform as a spatial vector under rotations.

Finally, the all-spatial bracket becomesUse $M_{ab}={ϵ}_{abr}{J}_r$ in the double contraction with ${ϵ}_{iab}{ϵ}_{jcd}/4$. The epsilon contraction identity reduces it to $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. One can check the sign directly: $J_1=M_{23}$ and $J_2=M_{31}$ give $[J_1,J_2]=i{M}_{12}=i{J}_3$; cyclic permutations give the other nonzero brackets. The requested coefficients areThese are rotation and boost commutators with a fixed metric signature. The PDF does not explicitly specify the signature. Keeping its generator convention but choosing $\eta =\mathrm{diag}(-1,1,1,1)$ reverses every displayed algebra coefficient: the pairs become $(1,0),(0,-1),(-1,0)$. More generally, if $s={\eta }_{00}=\pm 1$, the three nonzero coefficients are $A_1=-s$, $B_2=s$ and $A_3=s$. Specifying the Minkowski metric is therefore necessary to make the numerical signs unambiguous.