Remark

Tag 011X, permanent. Cite it as [Bourbaki, Tag 011X].

Lie Groups and Lie Algebras, chapter VIII, § 5. Automorphisms of a semi-simple Lie algebra

Let $\varepsilon',\varepsilon''$ be the restrictions of $\varepsilon$ to $\operatorname{Aut}_0(\mathfrak g,\mathfrak h)$ , $\operatorname{Aut}_e(\mathfrak g,\mathfrak h)$ . Let $f'$ be the homomorphism from $T_P$ to $\operatorname{Aut}_e(\mathfrak g,\mathfrak h)$ induced by $f$ via the canonical injection from $Q(R)$ to $P(R)$ . In the preceding we have established the following commutative diagram:

\begin{array}{ccccccccc} 1&\longrightarrow&T_Q&\xrightarrow{\ f\ }&\operatorname{Aut}(\mathfrak g,\mathfrak h)&\xrightarrow{\ \varepsilon\ }&A(\mathbf R)&\longrightarrow&1\\ &&\uparrow&&\uparrow&&\uparrow&&\\ 1&\longrightarrow&T_Q&\xrightarrow{\ f\ }&\operatorname{Aut}_0(\mathfrak g,\mathfrak h)&\xrightarrow{\ \varepsilon'\ }&W(\mathbf R)&\longrightarrow&1\\ &&\uparrow^{q}&&\uparrow&&\uparrow&&\\ &&T_P&\xrightarrow{\ f'\ }&\operatorname{Aut}_e(\mathfrak g,\mathfrak h)&\xrightarrow{\ \varepsilon''\ }&W(\mathbf R)&\longrightarrow&1 \end{array}

in which the vertical arrows other than

q

denote the canonical injections. We have seen (Prop. 2 and 4) that the first two rows are exact. In the third row, the homomorphism

\varepsilon''

is surjective (§2, no. 2, Cor. of Th. 2); it can be shown that its kernel is

f'(T_P)

(§7, Exerc. 26 d)).

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.

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