 Killing form of the SU(2) Lie algebra

In the anti-Hermitian basis $T^a=-i{\sigma }^a/2$, the SU(2) Lie algebra has $[T^a,T^b]={\epsilon }^{abc}{T}^c$ and Killing form $\kappa (T^a,T^b)=-2{\delta }^{ab}$. This is negative definite; the positive internal form for compact Yang-Mills theory is its negative.