Lemma 2

Tag 00VV, permanent. Cite it as [Bourbaki, Tag 00VV].

Lie Groups and Lie Algebras, chapter VII, § 4. Regular elements of a Lie group

Let $0\rightarrow V'\rightarrow V\rightarrow V''\rightarrow 0$ be an exact sequence of G-modules defined by analytic linear representations $\rho ', \rho , \rho ''$ of G, respectively. Then:

$r_{\rho}=r_{\rho'}+r_{\rho''}$ , and $r^0_{\rho}=r^0_{\rho'}+r^0_{\rho''}$ .

Indeed, for all $g\in G$ , there is (§1, no. 1, Cor. 3 of Th.1 ) an exact sequence

0\rightarrow (V')^1(\rho '(g))\rightarrow V^1(\rho (g))\rightarrow (V'')^1(\rho ''(g))\rightarrow 0

which proves the first assertion. The second follows from it since, by Lemma 1 (iv), $r^0_{\rho}=r_{\rho}, r^0_{\rho'}=r_{\rho'}$ and $r_{\rho''}^0=r_{\rho''}$ on a dense open subset of G.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

§1, no. 1, Cor. 3 of Th.1
Lemma 1