= Uniqueness of the two-derivative two-flavour chiral action
{title2=$\mathcal L_2=F^2\operatorname{tr}(\partial_\mu U^\dagger\partial^\mu U)/4$}
For $U\in SU(2)$ with $U\mapsto AUB^{-1}$, the chiral group acts transitively. A derivative-free invariant is therefore constant. At $U=I$, its diagonal stabilizer acts on tangent vectors by the irreducible three-dimensional rotation representation; the only invariant symmetric metric is a multiple of $\delta_{ab}$. Transporting this metric gives the displayed unique Lorentz-invariant two-derivative action up to normalization and total derivatives. Equivalently, $U^\dagger\partial_\mu U$ is traceless, eliminating a separate double-trace invariant. Writing $U=\exp(i\pi_a\tau_a/F)$ and using $\operatorname{tr}(\tau_a\tau_b)=2\delta_{ab}$ gives canonically normalized <pion> kinetic terms.
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