Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 2 / d / Solution

Use ${ϵ}_{xyz}=+1$ and the supplied commutators. Antisymmetry also gives $[K_a,L_b]={ϵ}_{abc}{K}_c$. Expanding the combinations,The mixed commutator cancels:ThusThere are two mutually commuting three-generator rotation algebras. As real compact Lie algebras,For example, the anti-Hermitian basis $-i{\sigma }_a/2$ obeys the same commutation convention in each $\mathfrak{su}(2)$ factor. This is an algebra statement. Globally, the simply connected group is $\mathrm{Spin}(4)\cong SU(2)\times SU(2)$, andDropping that central quotient would incorrectly identify the global groups.