rt.representation theory - What's the origin of the naming convention for the standard basis of sl_2?

Both terminology and notation in Lie theory have varied over time, but as far
as I know the letter H comes up naturally (in various fonts) as the next
letter after G in the early development of Lie groups. Lie algebras came
later, being viewed initially as "infinitesimal groups" and having labels like
those of the groups but in lower case Fraktur. Many traditional names are
not quite right: "Cartan subalgebras" and such arose in work of Killing,
while the "Killing form" seems due to Cartan (as explained by Armand Borel,
who accidentally introduced the terminology). It would take a lot of work to
track the history of all these things. In his book, Thomas Hawkins is more
concerned about the history of ideas. Anyway, both (h,e,f) and (h,x,y) are
widely used for the standard basis of a 3-dimensional simple Lie algebra,
but I don't know where either of these first occurred. Certainly h belongs to
a Cartan subalgebra.

My own unanswered question along these lines is the source of the now
standard lower case Greek letter rho for the half-sum of positive
roots (or sum of fundamental weights). There was some competition from
delta, but other kinds of symbols were also used by Weyl, Harish-Chandra, ....

ADDED: Looking back as far as Weyl's four-part paper in Mathematische Zeitschrift (1925-1926), but not as far back as E. Cartan's Paris thesis, I can see clearly in part III the prominence of the infinitesimal "subgroup" $mathfrak{h}$ in the structure theory of infinitesimal groups which he laid out there following Lie, Engel, Cartan. (Part IV treats his character formula
using integration over what we would call the compact real form of the semisimple Lie group in the background. But part III covers essentially the Lie algebra structure.) The development starts with a solvable subgroup $mathfrak{h}$ and its "roots" in a Fitting decomposition of a general Lie algebra, followed by Cartan's criterion for semisimplicity and then the more familiar root space decomposition. Roots start out more generally as "roots" of the characteristic polynomial of a "regular" element. Jacobson follows a similar line in his 1962 book, reflecting his early visit at IAS and the lecture notes he wrote there inspired by Weyl.

In Weyl you also see an equation of the type $[h e_alpha] = alpha cdot e_alpha$, though his notation is quite variable in different places and sometimes favors lower case, sometimes upper case for similar objects. Early in the papers you see an infinitesimal group $mathfrak{a}$ with subgroup $mathfrak{b}$.
All of which confirms my original view that H is just the letter of the alphabet following G, as often encountered in modern group theory. (No mystery.)