= Solution
Fix the <Minkowski metric> convention $\eta=\operatorname{diag}(1,-1,-1,-1)$ and the <Levi-Civita symbol> $\epsilon_{123}=1$. The antisymmetry of the <Lorentz algebra> generators gives $M_{i0}=-K_i$ and the inverse relation $M_{ij}=\epsilon_{ijk}J_k$. Inserting one temporal index in each generator immediately yields
$$
[K_i,K_j]=-i\eta_{00}M_{ij}=-i\epsilon_{ijk}J_k.
$$
Two <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> gives
$$
[M_{ab},M_{0j}]=i(\delta_{aj}K_b-\delta_{bj}K_a).
$$
Contracting with $\epsilon_{iab}/2$ gives
$$
[J_i,K_j]=\frac i2\epsilon_{iab}(\delta_{aj}K_b-\delta_{bj}K_a)
=i\epsilon_{ijk}K_k.
$$
Thus the three <Lorentz boosts> transform as a spatial vector under rotations.
Finally, the all-spatial bracket becomes
$$
[M_{ab},M_{cd}]
=i(-\delta_{bc}M_{ad}-\delta_{ad}M_{bc}+\delta_{bd}M_{ac}+\delta_{ac}M_{bd}).
$$
Use $M_{ab}=\epsilon_{abr}J_r$ in the double contraction with $\epsilon_{iab}\epsilon_{jcd}/4$. The epsilon contraction identity reduces it to $[J_i,J_j]=i\epsilon_{ijk}J_k$. One can check the sign directly: $J_1=M_{23}$ and $J_2=M_{31}$ give $[J_1,J_2]=iM_{12}=iJ_3$; cyclic permutations give the other nonzero brackets. The requested coefficients are
$$
\boxed{(A_1,B_1)=(-1,0),\qquad(A_2,B_2)=(0,1),\qquad(A_3,B_3)=(1,0).}
$$
These are <rotation and boost commutators with a fixed metric signature>. The PDF does not explicitly specify the signature. Keeping its generator convention but choosing $\eta=\operatorname{diag}(-1,1,1,1)$ reverses every displayed algebra coefficient: the pairs become $(1,0),(0,-1),(-1,0)$. More generally, if $s=\eta_{00}=\pm1$, 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.
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