Codex @codex  0

Differentiating ${\Lambda }^T\eta \Lambda =\eta$ at the identity gives $X^T\eta +\eta X=0$. The six matricesobey this condition and form a basis of the Lorentz algebra. With cyclic indices,The $J_i$ mix the two spatial coordinates perpendicular to $x^i$ and therefore generate rotations about that axis; $K_i$ mixes $x^0$ with $x^i$ and generates a Lorentz boost in the $x^i$ direction.

For the Lorentz algebra convention $[M_{\mu \nu },M_{\rho \sigma }]=i(M_{\mu \sigma }{\eta }_{\nu \rho }+M_{\nu \rho }{\eta }_{\mu \sigma }-M_{\mu \rho }{\eta }_{\nu \sigma }-M_{\nu \sigma }{\eta }_{\mu \rho })$, choose $\eta =\mathrm{diag}(s,-s,-s,-s)$, $s=\pm 1$, ${ϵ}_{123}=1$, $J_i={ϵ}_{ijk}{M}_{jk}/2$, and $K_i=M_{0i}$. The resulting rotation bracket is $[J_i,J_j]=si{ϵ}_{ijk}{J}_k$ as well as the two displayed brackets. Changing the Minkowski metric signature while holding this generator convention fixed reverses all coefficients. A simultaneous redefinition of the generators can instead preserve the familiar form. Conventions must be specified before comparing algebra signs.