Rather than prolong the tangled comments, I'll try to provide a straightforward answer to the current formulation of the question. As noted already, there are two small cases of irreducible root systems (belonging to simple Lie algebras) in which the dominant (open) Weyl chamber denoted $K$ here contains a root: type $A_1$ and $2\rho = \alpha$ (the sole positive root), type $A_2$ and $\rho = \alpha_1 + \alpha_2$.

Beyond these cases, there are no others. Possibly there is a somewhat more conceptual proof of this, but at the moment I can only verify case-by-case that the highest root $\tilde{\alpha}$ and (if there are two root lengths) the highest short root $\alpha_0$ lie in the closure of $K$ but not in $K$ itself. In the background is the fact that $\tilde{\alpha}$ and (if it exists) $\alpha_0$ are the only roots which are *dominant* weights: highest weights of irreducible finite dimensional representations. (This fact does have conceptual origins, which I won't go into here.)

In order to study the root systems individually, it's best to rely on the tables at the end of Bourbaki's Chapters 4-6 in *Lie Groups and Lie Algebras*. For the highest short roots, when they exist, it may be easiest to look at the table in my textbook on page 66. For the exceptional types, it's useful to refer to the explicit tables of positive roots arranged by height appended to Springer's IHES paper here. In all cases, what has to be checked is that some simple root is orthogonal to the dominant root in question, i.e., adding or subtracting it doesn't produce another root. (The dominance means that the addition part is automatic.)

ADDED: After all the edits and comments, maybe it's useful to outline a more detailed proof (based on the Bourbaki/Serre treatment of root systems). Note that only the axiomatic theory of root systems is actually involved here, leaving aside the origins in Lie algebras.

While developing basic facts about a (reduced) root system in a real euclidean space $E$, with Weyl group $W$, one needs to show that there exist simple systems of roots $\alpha_1, \dots, \alpha_n$ and resulting positive systems (all conjugate under $W$). In particular, $W$ is generated by the reflections $s_i$ corresponding to the $\alpha_i$, and the positive roots have a natural partial ordering $\beta \leq \alpha$.

The proofs show that in fact $W$ acts simply transitively on the systems of simple roots. When all hyperplanes orthogonal to roots are removed from $E$, the remaining open sets (Weyl chambers) are then permuted simply transitively by $W$. One of these is the positive (or dominant) Weyl chamber denoted here by $K$, defined by the condition that $(\lambda, \alpha_i) > 0$ for all $\lambda \in K$ and all $\alpha_i$. Then one shows that the closure $\overline{K}$ is a *fundamental domain* for $W$: every element of $E$ is $W$-conjugate to a unique element of $\overline{K}$.

With this information in place, one can say more special things about an *irreducible* root system (one coming from a simple Lie algebra):

1) Under the partial ordering, there is a unique maximal root $\tilde{\alpha}$.

2) $W$ acts irreducibly on $E$.

3) There are at most two root lengths (long, short), those of the same length being $W$-conjugate (so only one of a given length can lie in $\overline{K}$). (If all roots are of equal length, the system is usually called "simply-laced".)

4) If there are two root lengths, $\tilde{\alpha}$ is long. Comparing with the *dual* root system (where long and short are interchanged), one sees that in this case there is a unique maximal short root $\alpha_0$ in the partial ordering of the original system. Moreover, $\alpha_0$ lies in $\overline{K}$.

Only at this point does the *classification* of irreducible root systems by Dynkin diagrams come into play. As I've indicated above, one can identify case-by-case which highest long/short roots are orthogonal to particular simple roots; as noted, the only cases where $\tilde{\alpha} \in K$ are types $A_1, A_2$.

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