Two-dimensional Killing algebra with a periodic generator is abelian ($[T,\Phi ]=0$)

Suppose the full Lie algebra of Killing fields has basis $T,\Phi$ and the flow of $\Phi$ has a nonzero period $P$. Write $[T,\Phi ]=aT+b\Phi$ with constant coefficients. The Adjoint representation of the periodic flow satisfies $e^{P\mathrm{ad}\Phi }=1$. Its real eigenvalues are $-a$ and zero, so $a=0$. The remaining adjoint operator is nilpotent and its exponential is $1+P\mathrm{ad}\Phi$, forcing $b=0$. This proves commutation from genuine periodic axial symmetry, without differentiating mere asymptotic component limits.