The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in Bourbaki's language). Here there are at most two possible lengths of roots, relative to a euclidean metric arising from the nondegenerate Killing form restricted to a fixed but arbitrary Cartan subalgebra. In the "simply-laced" types ADE, all roots have equal length, whereas in other types BCFG there are both "long" and "short" roots.

At least since the early work of Dynkin, it has been conventional to say that all roots in types ADE are "long" (though it would be possible to say all are "short", or else just avoid the distinction here). For example, Dynkin's method for drawing his diagrams uses open circles for long simple roots, filled-in circles for short simple roots.

Is there a most natural rationale for the convention that all roots are long in types ADE?

One situation in which the convention seems handy involves the classification of the finitely many nilpotent orbits in the Lie algebra (which also goes back to Dynkin). Here one finds a unique minimal nonzero orbit, characterized as the set of root vectors corresponding to *long* roots (for any choice of Cartan subalgebra). But I'm uncertain about the main motivation behind the convention (and its history).

ADDED: I was thinking at first just about the algebraic theory of simple Lie algebras, which can be studied over any splitting field of characteristic 0 in terms of root systems and Weyl groups. Over $\mathbb{C}$ these arise classically as complexifications of various real Lie algebras belonging to Lie groups. In this direction the answer (resp. comment) by Andre (resp. Henrik) are both useful. The paper by Jeff Adams is a unification of previous case-by-case study, working in a traditional Lie group setting, whereas the KNR paper also brings in some more modern ideas as well as tools from algebraic geometry. I can accept such an answer but should wait a little longer in case someone else can suggest an answer in the purely algebraic setting I've sketched. (By the way, in his papers from about 1946 to 1950, Dynkin's convention for labelling vertices seems to have evolved. Maybe this was influenced by his classification of nilpotent orbits, including the description of the minimal nonzero orbit?)

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