Proposition 9

Let $k$ be a field of characteristic 0 and $\mathfrak{g}$ a Lie algebra over $k$. If $a\in$ Aut$_e(\mathfrak{g})$, the dimension of the nilspace of $a-1$ is greater than or equal to the rank of $\mathfrak{g}$.

By the “Lefschetz principle” (Algebra, Chap. V, §14, no. 6, Cor. 2 of Th. $5$)$,k$ is an ascending directed union of subfields $(k_i)_{i\in I}$ which admit $\mathbf{C}$ as extension field. Let $(e_{\alpha})$ be a basis of $\mathfrak{g}$ over $k$ and $x_1, . . . , x_m$ elements of $\mathfrak{g}$ such that ad($x_1$)$, . . .$, ad($x_m$) are nilpotent and $a=e^{ad(x_1)}. . . e^{ad(x_m)}$. Let $c^{\gamma}_{\alpha \beta}$ be the structure constants of $\mathfrak{g}$ with respect to the basis $(e_{\alpha})$ and $(x^{\alpha}_r)$ the components of $x_r$ with respect to this basis $(1\leq r\leq m)$. There exists an index $j\in I$ such that the $c^{\gamma}_{\alpha \beta}$ and the $x^{\alpha}_r$ all belong to $k_j$. Let $\mathfrak{g}_j=\sum_{\alpha}k_je_{\alpha}$;

this is a Lie algebra over $k_j$ containing $x_1, . . . , x_m$, and the restriction $a_j$ of $a$ to $\mathfrak{g}_j$ is an elementary automorphism of $\mathfrak{g}_j$. The extension of $a_j$ to $\mathfrak{g}_j\otimes_{k_j}\mathbf{C}$ is an elementary automorphism $a_j\otimes 1$ of $\mathfrak{g}_j\otimes \mathbf{C}$. So let $G_j$ be a connected complex Lie group with Lie algebra $\mathfrak{g}_j\otimes \mathbf{C}$, and $s$ an element of $G_j$ such that Ad($s$) $=a_j\otimes 1$. Prop. 8, applied to the pair $(G_j, s)$, shows that the nilspace of $a_j\otimes 1-1$ is of dimension $n$, so

$n\geq$ rk($\mathfrak{g}_j\otimes \mathbf{C}$) $=$ rk($\mathfrak{g}_j$) $=$ rk($\mathfrak{g}$).

But this nilspace has the same dimension as that of $a_j-1$ and that of $a-1$. Hence the proposition.