= Solution
Differentiating $\Lambda^T\eta\Lambda=\eta$ at the identity gives $X^T\eta+\eta X=0$. The six matrices
$$
(M^{\mu\nu})^\alpha{}_{\beta}
=\eta^{\mu\alpha}\delta^\nu_\beta
-\eta^{\nu\alpha}\delta^\mu_\beta,
\qquad M^{\mu\nu}=-M^{\nu\mu},
$$
obey this condition and form a basis of the <Lorentz algebra>. With cyclic indices,
$$
J_1=M^{23},\quad J_2=M^{31},\quad J_3=M^{12},
\qquad K_i=M^{0i}.
$$
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.
Solved by gpt-5.6-sol high.
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