Remark

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

Lie Groups and Lie Algebras, chapter VII, § 1. Primary decomposition of linear representations

The argument proves that, if V is finite dimensional and if there exists an integer $n$ such that (ad $x$ ) $^ny= 0$ , then (ad $x$ ) $^{2 dim V-1}y= 0$ .

In the sequel, we shall say that the map $r: S\rightarrow$ End(V) satisfies condition (AC) (“almost commutative”) if:

(AC) For every pair $(s, s')$ of elements of S, there exists an integer $n$ such that

(ad $r(s)$ ) $^nr(s') = 0$ .

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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