Lorentz algebra in three spacetime dimensions ($\mathfrak{so}(1,2{)}_{\mathbb{C}}\cong {\mathfrak{sl}}_2(\mathbb{C})$)

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 $m$.