Solution
= Solution
The Lie <derivative> formulation is $\mathcal L_Vg=0$. Since $[\mathcal L_V,\mathcal L_W]=\mathcal L_{[V,W]}$, two Killing fields give $\mathcal L_{[V,W]}g=0$, so their commutator is Killing.
= Solution
The Lie <derivative> formulation is $\mathcal L_Vg=0$. Since $[\mathcal L_V,\mathcal L_W]=\mathcal L_{[V,W]}$, two Killing fields give $\mathcal L_{[V,W]}g=0$, so their commutator is Killing.