Lemma 1

Let V be a finite dimensional vector space, $\mathfrak{n}$ a Lie subalgebra of $\mathfrak{a}=\mathfrak{g}\mathfrak{l}(V)$ consisting of nilpotent elements.

(i) The map $x \rightarrow$ exp $x$ is a bijection from $\mathfrak{n}$ to a subgroup N of $\mathbf{G}\mathbf{L}(V)$ consisting of unipotent elements (Chap. II, §6, no. 1, Remark 4). We have $\mathfrak{n}=$ log(exp$\mathfrak{n}$). The map $f \rightarrow f\circ$ log is an isomorphism from the algebra of polynomial functions on $\mathfrak{n}$ to the algebra of restrictions to N of polynomial functions on End(V).

(ii) If $x\in \mathfrak{n}$ and $a\in \mathfrak{a}$,

(exp ad$_{\mathfrak{a}}x$)$.a=$ (exp $x$)$a$(exp($-x$)).

(iii) Let $V'$ be a finite dimensional vector space, $\mathfrak{n}'$ a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V')$ consisting of nilpotent elements, $\rho$ a homomorphism from $\mathfrak{n}$ to $\mathfrak{n}'$. Let $\pi$ be the map exp $x \rightarrow$ exp $\rho (x)$ from exp$\mathfrak{n}$ to exp$\mathfrak{n}'$. Then $\pi$ is a group homomorphism.

By Engel’s theorem, we can identify V with $k^n$ in such a way that $\mathfrak{n}$ is a subalgebra of $\mathfrak{n}(n, k)$ (the Lie subalgebra of $\mathbf{M}_n(k)$ consisting of the lower triangular matrices with zeros on the diagonal). For $s\geq 0$, let $\mathfrak{n}_s(n, k)$ be the set of $(x_{ij})_{1\leq i,j\leq n}\in \mathbf{M}_n(k)$ such that $x_{ij}= 0$ for $i-j < s$. Then

(Chap. II, §4, no. 6, Remark), and the Hausdorff series defines a polynomial map $(a, b) \rightarrow H(a, b)$ from $\mathfrak{n}(n, k)\times \mathfrak{n}(n, k)$ to $\mathfrak{n}(n, k)$ (Chap. II, §6, no. 5, Remark 3); this map makes $\mathfrak{n}(n, k)$ into a group (Chap. II, §6, no. 5, Prop. 4). By Chap. II, §6, no. 1, Remark 4, the maps $x \rightarrow$ exp $x$ from $\mathfrak{n}(n, k)$ to $1 +\mathfrak{n}(n, k)$ and $y \rightarrow$ log $y$ from $1 +\mathfrak{n}(n, k)$ to $\mathfrak{n}(n, k)$ are inverse bijections and are polynomial; by Chap. II, §6, no. 5, Prop. 3, these maps are isomorphisms of groups if $\mathfrak{n}(n, k)$ is given the group law $(a, b) \rightarrow H(a, b)$ and if $1 +\mathfrak{n}(n, k)$ is considered as a subgroup of $\mathbf{G}\mathbf{L}_n(k)$. Assertions (i) and (iii) of the lemma now follow. Let $x\in \mathfrak{n}$. Denote by $L_x,R_x$ the maps $u \rightarrow xu, u \rightarrow ux$ from $\mathfrak{a}$ to $\mathfrak{a}$, which commute and are nilpotent. We have ad$_{\mathfrak{a}}x= L_x-R_x$, so, for all $a\in \mathfrak{a}$,

(exp ad$_{\mathfrak{a}}x$)$a=$ (exp(L$_x-R_x$))$a=$ (exp $L_x$)(exp $R_{-x}$)$a$ (1)

$=\sum Li^{ix}$! $Rj\overset{j-x}{!}a=$ (exp $x$)$a$(exp($-x$)).

$i,j\geq 0$

With the notation in Lemma $1,\pi$ is called the linear representation of exp $\mathfrak{n}$ compatible with the given representation $\rho$ of $\mathfrak{n}$ on $V'$. When $k$ is $\mathbf{R}$, $\mathbf{C}$, or a non-discrete complete ultrametric field, $\rho = L(\pi )$ by the properties of exponential maps (Chap. III, §4, no. 4, Cor. 2 of Prop. 8).