= Two-dimensional Killing algebra with a periodic generator is abelian
{title2=$[T,\Phi]=0$}
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 $e^{P\operatorname{ad}\Phi}=1$. Its real eigenvalues are $-a$ and zero, so $a=0$. The remaining adjoint operator is nilpotent and its exponential is $1+P\operatorname{ad}\Phi$, forcing $b=0$. This proves commutation from genuine periodic axial symmetry, without differentiating mere asymptotic component limits.
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