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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 2 / a / Solution

{c}
{title2=$\mathfrak{so}(1,3)$}
{wiki=Lorentz_group#The_Lie_algebra}

The Lorentz algebra has rotation generators $J_i$ and boost generators $K_i$ satisfying
$$
[J_i,J_j]=\epsilon_{ijk}J_k,\qquad
[J_i,K_j]=\epsilon_{ijk}K_k,\qquad
[K_i,K_j]=-\epsilon_{ijk}J_k.
$$


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