Lemma 1

Let A be an associative algebra over $k,H$ and $X$ elements of A such that $[H, X] = 2X$.

(i) $[H, X^n] = 2nX^n$ for any integer $n\geq 0$.

(ii) If $Z$ is an element of A such that $[Z, X] =H$, then, for any integer $n >0$,

The map $T \rightarrow [H, T]$ from A to A is a derivation, which implies (i). With the assumptions in (ii),

On the other hand, $X^{n-1}(H+n-1) = (H-n+ 1)X^{n-1}$ by (i). Q.E.D.

Recall that we denote by $\mathfrak{s}\mathfrak{l}(2, k)$ the Lie algebra consisting of the square matrices of order 2, trace zero, and with entries in $k$. This Lie algebra is simple of dimension 3 (Chap. I, §6, no. 7, Example). The canonical basis of $\mathfrak{s}\mathfrak{l}(2, k)$ is the basis $(X_+, X_-, H)$, where

$(01)($ 0 $0)(1$ 0 $)$

$X_+=$ 0 0 $X_-=-1$ 0 $H=$ 0 $-1$.

We have

Since the identity representation of $\mathfrak{s}\mathfrak{l}(2, k)$ is injective, $H$ is a semi-simple element of $\mathfrak{s}\mathfrak{l}(2, k)$ and $X_+, X_-$ are nilpotent elements of $\mathfrak{s}\mathfrak{l}(2, k)$ (Chap. I, §6, no. 3, Th. 3). By Chap. VII, §2, no. 1, Example $4,kH$ is a Cartan subalgebra of $\mathfrak{s}\mathfrak{l}(2, k)$. The map $U \rightarrow -^tU$ is an involutive automorphism of the Lie algebra $\mathfrak{s}\mathfrak{l}(2, k)$, called the canonical involution of $\mathfrak{s}\mathfrak{l}(2, k)$; it transforms $(X_+, X_-, H)$ into $(X_-, X_+,-H)$.