One of the basic results of Lie theory is that if one picks a Cartan subalgebra of a simple Lie algebra, there there is a canonical decomposition of $\mathfrak{g}$ into the Cartan and a bunch of 1-dimensional subspaces (root spaces). Picking any basis of the Cartan (I'm not going to worry about which one), one is thus extremely close to having a canonically chosen basis of the Lie algebra, except that there are a bunch of scalar factors running around. I'd like to fix a particular choice of basis, while making a somewhat manageable number of choices.

Choice 1: I'll pick a base of my root system, that is, a set of simple roots/positive roots/a Borel containing my chosen Cartan. I can now choose completely at random vectors $E_i$ in the simple root spaces (all of these are conjugate by inner automorphisms, so they are all "the same"). This fixes basis vectors $F_i$ in the negative simple root spaces that commute with the $E_i$'s as the standard generators of $\mathfrak{sl}_2$.

Choice 2: Well, now I have to do something else non-canonical. What I'd like to choose is a convex order on my positive roots. This case, let be propose a basis for the other root spaces by $E_{\alpha+\beta}=[E_{\alpha},E_{\beta}]$ if $\alpha <\beta$ and similarly for the $F$'s.

My question: is $E_{\alpha}$ uniquely defined? Has this construction appeared before in the literature? If this definition doesn't work, is there a variant on it which does?