Lie algebra of the Lorentz group ($\mathfrak{o}(1,3)$)

For $M=\mathrm{diag}(-1,1,1,1)$,Every such matrix has the block formso the Lie algebra and the Lorentz group have dimension six. At $A\in O(1,3)$, the tangent space is $A\mathfrak{o}(1,3)$.

With signature $(-,+,+)$, two Lorentz boost generators and one spatial rotation obey $[J,K_1]=K_2$, $[J,K_2]=-K_1$ and $[K_1,K_2]=-J$. Replacing the boosts by $i{K}_1,i{K}_2$ gives the compact cyclic brackets, proving the complexified isomorphism with the SU(2) Lie algebra. The real algebra has dimension three and rank one. Its finite-dimensional complex irreducible representations have dimensions $m+1$ for nonnegative integer highest weight $m$.