In the main text we showed that gravitational and gauge anomalies cancel in 10 dimensions for N = 1 supergravity coupled to Yang–Mills theory if the gauge group G is either SO(32) or E8 × E8. None of the other “classical groups” (SO(n), SU(n) and Sp(n)) was allowed. We now complete this analysis by discussing the exceptional groups, namely G2, F4, E6, E7 and E8. As we have shown in the main text, cancellation of gravitational anomalies allows only Lie groups with 496 generators. There are clearly many products of simple Lie algebras with this number of generators. In particular, there are semisimple Lie algebras with one or more exceptional groups as simple factors. However, we can at once rule out these exceptional groups if we study one-loop hexagon graphs with six gauge fields all belonging to the same exceptional Lie group, and if factorization of the kind discussed in the main text does not occur. In four dimensions gauge anomalies are proportional to the symmetrized trace of three generators, dabc(R) = tr(Ta{Tb, Tc}), where Ta are the generators of the gauge group in a representation R, and thus real or pseudoreal representations do not carry anomalies. A representation R can only carry an anomaly if dabc(R) is nonvanishing, and this is only possible if there exists a cubic Casimir operator for the group.