Remark

Tag 0A4O, permanent. Cite it as [Bourbaki, Tag 0A4O].

Lie Groups and Lie Algebras, chapter II, § 4. Central filtrations

Let $\alpha \geq 0, \beta \geq 0$ and $x \in \Gamma_\alpha, y \in \Gamma_\beta$ . By formulae (21), (22) and (23),

(24) \quad x^{-1} - 1 \equiv -(x - 1) \pmod{A_{2\alpha}}

(25) \quad xy - 1 \equiv (x - 1) + (y - 1) \pmod{A_{\alpha + \beta}}

(26) \quad (x, y) - 1 \equiv [(x - 1), (y - 1)] \pmod{A_{\alpha + \beta + \inf(\alpha, \beta)}}.

We prove for example (26). If $x - 1 = \xi$ and $y - 1 = \eta$ , (23) gives:

(x, y) - 1 - [\xi, \eta] = ((x^{-1} - 1) + (y^{-1} - 1) + (x^{-1} - 1)(y^{-1} - 1))[\xi, \eta].

Now

[\xi, \eta] \in A_{\alpha + \beta}

(x^{-1} - 1) \in A_\alpha

(y^{-1} - 1) \in A_\beta

, whence we obtain (26).

Let $G$ be a group and $\rho : G \to \Gamma$ a homomorphism. For all real $\alpha$ , we write $G_\alpha = \rho^{-1}(\Gamma_\alpha)$ . As $(\Gamma_\alpha)$ is a central filtration on $\Gamma$ , it is immediate that $(G_\alpha)$ is a central filtration on $G$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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