Relation Between Triangular and Witten SU(2) Anomaly Cancellation for Gauge Groups

We show by group-theoretic means that the absence of the Witten SU(2) (global) anomaly for gauge groups is guaranteed by the (perturbative) triangular anomaly-free condition as long as the SU(2) gauge group is embedded in a larger simple gauge group $G$ with the property ${\ensuremath{\Pi}}_{4}(G)=0$ (${\ensuremath{\Pi}}_{4}$ is the four-dimensional homotopy group). The inverse is not true. The same method could be used to relate perturbative and global anomaly cancellation in more than four dimensions.