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(This at any rate is the kind of answer given to the terminology question raised with Mostow at a conference I attended decades ago.) …

Mostow's theorem actually has a relatively modern textbook treatment, by his early collaborator Gerhard Hochschild: see Theorem 4.3 in VIII.4 of his book Basic Theory of Algebraic Groups and Lie Algeb …

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 … 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 …

I'm less familiar with terminology in the theory of Lie rings and abstract nilpotent groups. …

I think the answer to the original question is that there is no special name for the involution (otherwise it would have turned up by now). My first encounter with it was in the 1979 Kazhdan-Lusztig …

Many of the examples mentioned go back to earlier centuries, when insulated national traditions and slow communications promoted mistaken labelling of results and concepts. A much more recent exampl …

Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine pa …

While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into …

Your basic question about a reference does go back to Weyl's complete reducibility theorem (I'd have to check his book on classical groups, but it isn't just a result about classical Lie groups). F …

The correct answer to the narrow question asked about terminology has been given by quid, but I'm tempted to add some broader semi-historical remarks as well. … As usual, the development of mathematical notation and terminology is somewhat arbitrary but complicated. …

Marty is definitely correct about the origin of the terminology in the study of homogeneous spaces $G/H$ of reductive Lie groups. …

In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A} …

Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's tre …

It's unclear to me why this terminology is so popular, since over a field of characteristic 0 it's equivalent to "reductive algebraic group". …

Terminology in mathematics develops a bit haphazardly, and sometimes things get misleading names. … Anyway, one might make a case for the terminology "James module" here, but it's too late for that. Specht himself had no special influence on the modular theory. …