Proposition 8

Tag 00U8, permanent. Cite it as [Bourbaki, Tag 00U8].

Lie Groups and Lie Algebras, chapter VII, § 2. Cartan subalgebras and regular elements of a Lie algebra

Let $f:\mathfrak{g}\rightarrow \mathfrak{g}'$ be a surjective homomorphism of Lie algebras.

(i) If $x$ is a regular element of $\mathfrak{g},f(x)$ is regular in $\mathfrak{g}'$ . The converse is true if Ker $f$ is contained in the centre of $\mathfrak{g}$ .

(ii) rk( $\mathfrak{g}$ ) $\geq$ rk( $\mathfrak{g}'$ ).

Put rk( $\mathfrak{g}$ ) $=r$ , rk( $\mathfrak{g}'$ ) $=r'$ . Let $x\in \mathfrak{g}$ . The characteristic polynomials of ad $x$ , ad $f(x)$ and ad $x|$ Ker $f$ are of the form

P(T) = T^n+a_{n-1}(x)T^{n-1}+\cdots +a_r(x)T^r

Q(T) = T^{n'}+b_{n'-1}(x)T^{n'-1}+\cdots +b_{r'}(x)T^{r'}

R(T) = T^{n''}+c_{n''-1}(x)T^{n''-1}+\cdots +c_{r''}(x)T^{r''}

where the $a_i, b_i, c_i$ are polynomial functions on $\mathfrak{g}$ , with $a_r\not= 0, b_{r'}\not= 0, c_{r''}\not= 0$ . We have P = QR, so $r=r'+r''$ and $a_r(x) =b_{r'}(x)c_{r''}(x)$ , which proves (ii) and the first assertion of (i). If Ker $f$ is contained in the centre of $\mathfrak{g}$ , $R(T) = T^{n''}$ and so $a_r(x) =b_{r'}(x)$ , hence the second assertion of (i).

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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