I'm reading "lie algebras and particle physics" by Georgi and on I'm up top where he is creating the simple algebras from simple roots and there is something I am not getting here. On page 108 he seems to be making the claim that any simple root φ had the property that any lowering operator of any SU(2) subalgebra containing it must annihilate it. I don't see why this must be the case. Let's assume it's false, then there exists roots α, β such that α+β =φ. If α, β are both negative, then this is impossible since φ being simple means it must be positive. If α, β are positive then φ isn't simple, which we assumed it was. But why can't we have α positive and β negative or vice versa? In fact I would naively imagine that that's exactly what would happen. Consider some spin j SU(2) subalgebra, I would imagine that root corresponding to spin -j would be negative and it would be "raised" up to positive roots and eventually end up at some positive weight corresponding to +j. Which would mean that any simple root, since it must be positive, cannot be a lowest state and consequentially any simple root can always be lowered. Am I wrong? Am I missing something?