 Two commuting rotation algebras in so4

If $[L_a,L_b]={ϵ}_{abc}{L}_c$, $[L_a,K_b]={ϵ}_{abc}{K}_c$ and $[K_a,K_b]={ϵ}_{abc}{L}_c$, then $L_a^{\pm }=(L_a\pm K_a)/2$ generate commuting copies of $\mathfrak{su}(2)$. This Lie-algebra decomposition does not remove the central quotient in the corresponding global group.