The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 theory of finite-dimensional representations of a semisimple Lie algebra. (This question was recently raised by a colleague, who had observed that it was true in type $A$ and was tempted to go on in case-by-case fashion.)


First fix a simple system of roots in an irreducible root system, hence notions of *positive* roots and *dominant* weights. Recall the usual partial ordering of weights: $\mu \leq \lambda$ means that $\lambda -\mu$ is a sum of positive roots. There is some evidence that the following statement is always true:


$(\*)$ Suppose $\lambda, \mu$ are dominant weights. If $\lambda > \mu$ and there is no intermediate dominant weight, then $\lambda - \mu$ is a single root (not necessarily simple, of course).



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> Is $(\*)$ true for all root systems, and if so is there a reference?
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[UPDATE] Jantzen's recent short paper [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=3710660) has an affirmative case-by-case answer in 2.1-2.2. Although his paper deals with modular representations of semisimple algebraic groups, this part of the proof only concerns root-and-weight computations. (However, he does rely on the classification of irreducible root systems, so it's still a question whether there is a uniform proof.)


P.S. The argument due to Steinberg in Stembridge's 1998 paper seems to answer this last question, as pointed out by Sam Hopkins. 



Difference of adjacent dominant weights is a root?
This is true. See "The partial order of dominant weights" by John Stembridge, 1998 (<https://www.sciencedirect.com/science/article/pii/S0001870898917364>). In particular look at Corollary 2.7. Alternatively, look at the second proof of Corollary 2.7 given there, due to Robert Steinberg, which the paper claims was communicated by one James Humphreys :)


