Sundarshan's proof of the spin-statistics theorem

Question

The proof starts with the $\rm SU(2)$ symmetric Lagrangian \begin{equation} \mathcal L=\sum_{r,s=1}^4\frac{1}{2}K_{rs}^0(\xi_r\dot\xi_s-\dot\xi_r\xi_s) \end{equation} where the fields $\xi$ either have integer spin (in which case $K_{rs}^0$ is an anti-symmetric matrix) or half-integer spin (in which case $K_{rs}^0$ is symmetric). The requirement $$[\xi_n,\delta S]=i\delta\xi_n$$ on the variation of the action gives the expression \begin{align*} i\delta\xi_n=\frac{1}{2}\int_\sigma d^3x \sum_{rs}K_{rs}^0(\xi_n\xi_r\delta\xi_s-\xi_n\delta\xi_r\xi_s-\xi_r\delta\xi_s\xi_n+\delta\xi_r\xi_s\xi_n). \end{align*} If $\xi$ is a commuting field, then we can write explicitly \begin{equation*} [\xi_n,\delta S]=\int_\sigma d^3x \sum_s\delta \xi_s(\mathbf x)\left[\xi_n(\mathbf y),\frac{1}{2}\sum_r(K_{rs}^0-K_{sr}^0)\xi_r(\mathbf x)\right]. \end{equation*} Here it is concluded that $K_{rs}^0$ must be anti-symmetric, and therefore the fields that commute are the same as the fields with integer spin. I don't understand this: why must $K_{rs}^0$ be anti-symmetric?