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\begin{document}

\( LL^{\top }=Id \) so a lorentz transformation can be thought of as a rotation in 4
dimensions.

infinitesimal transformation definition:

\( \delta x_{i}=\epsilon _{ijk}\delta \theta _{j}x_{k}=i\delta \theta _{i}M_{jik}x_{k} \).

This has been worked out for 3-d rotations.

\( M_{i}=i*\begin{array}{ccc}
0 & -\delta _{i3} & \delta _{i2}\\
\delta _{i3} & 0 & -\delta _{i1}\\
-\delta _{i2} & \delta _{i1} & 0
\end{array}
 \). With this, you can express \( R=e^{i\theta n_{i}M^{i}} \) where \( n_{i} \)is normalized. commutation
relations: \( [M_{i},M_{j}]=i\epsilon _{ijk}M_{k} \). In the lorentz case, \( L_{i}=Id+i\delta \beta _{i}K_{i} \) where \( K_{i}=-i*\begin{array}{cccc}
0 & \delta _{i1} & \delta _{i2} & \delta _{i3}\\
\delta _{i1} & 0 & 0 & 0\\
\delta _{i2} & 0 & 0 & 0\\
\delta _{i3} & 0 & 0 & 0
\end{array}
 \) so \( L=e^{i\beta _{i}K^{i}} \). \( [K_{i},K_{j}]=-i\epsilon _{ijk}M_{k} \). \( [M_{i},K_{j}]=i\epsilon _{ijk}K_{k} \)

\end{document}
