This is an old question, probably abandoned because its answer would require a short article (complete with references). While I can't supply such an article, I can point to some of Borel's writings which involve a definition of "real reductive group". In an old comment, Brian Conrad already referred to the added survey in Section 24C of Borel's second edition of *Linear Algebraic Groups* (GTM 126, Springer, 1991). There is also a set of his lecture notes intended for a 2003 Chinese summer school, published (along with other lectures) as: Lie groups and linear algebraic groups. I. Complex and real groups. *Lie groups and automorphic forms*, 1–49, AMS/IP Stud. Adv. Math., 37, Amer. Math. Soc.,
Providence, RI, 2006. (See Sections 5-6.)

The definition he uses is in some ways the simplest and most natural, I think. In the general framework of finite dimensional Lie algebras over an arbitrary field $K$ of characteristic 0, there are elementary definitions of *semisimple* and *reductive* Lie algebras. When you take $K = \mathbb{R}$ and work in the classical framework of real Lie groups, it's then natural to define a *connected* Lie group $G$ to be *reductive* if its Lie algebra is. Of course, the Lie algebra only sees local behavior, so one could leave it at that. However, disconnected Lie groups come up immediately when Lie group theory is combined with linear algebraic groups in the study of representations, automorphic forms, etc. So Borel adds in this case the extra requirement that $G$ have only finitely many connected components in the euclidean topology.

Where does this condition come from? Starting with a *connected* linear algebraic group (scheme) $H$ over $\mathbb{R}$, the resulting group $G:= H(\mathbb{R})$ is a real Lie group but need not be connected. An obvious example is the multiplicative group. But a basic theorem states that this Lie group has only finitely many components in the euclidean topology. The theorem comes, for instance, from Whitney's older work on real affine varieties but is also a consequence of a finiteness theorem in Galois cohomology proved in the work of Borel-Serre. (I'm not sure what the best modern proof of the theorem is, but that's another question.)

Borel's notes were intended partly to prepare for Wallach's related lectures. In any case, his approach is closely related to some of the other proposed definitions of real reductive group in the question. But it strikes me as a more straightforward starting point.

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