Uniqueness of the two-derivative two-flavour chiral action (${\mathcal{L}}_2=F^2\mathrm{tr}({\partial }_{\mu }{U}^{†}{\partial }^{\mu }U)/4$)

For $U\in SU(2)$ with $U\mapsto AU{B}^{-1}$, 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, $U^{†}{\partial }_{\mu }U$ is traceless, eliminating a separate double-trace invariant. Writing $U=\exp (i{\pi }_a{\tau }_a/F)$ and using $\mathrm{tr}({\tau }_a{\tau }_b)=2{\delta }_{ab}$ gives canonically normalized pion kinetic terms.