Sublattices generated by root differences

Let Φ   be an irreducible root system of exceptional type. In this paper we calculate the set of exponents of torsion subgroups of quotients ZΦ/LZΦ/L, where ZΦZΦ is the root lattice and L runs through the sublattices thereof which are generated by root differences. The maximum such exponent is of interest because it appears in the hypotheses of a result of Liebeck and Seitz concerning lifting of embeddings from finite groups of Lie type to algebraic groups.