Definition 1

A finite product of automorphisms of $\mathfrak{g}$ of the form $e^{adx}$ with ad $x$ nilpotent is called an elementary automorphism of $\mathfrak{g}$. The group of elementary automorphisms of $\mathfrak{g}$ is denoted by Aut$_e(\mathfrak{g})$.

If $u\in$ Aut($\mathfrak{g}$)$,ue^{adx}u^{-1}=e^{adu(x)}$. It follows that Aut$_e(\mathfrak{g})$ is a normal subgroup of Aut($\mathfrak{g}$). If $k=\mathbf{R}$ or $\mathbf{C}$, Aut$_e(\mathfrak{g})$ is contained in the group Int($\mathfrak{g}$) of inner automorphisms of $\mathfrak{g}$ (Chap. III, §6, no. 2, Def. 2).

\* In the general case, Aut$_e(\mathfrak{g})$ is contained in the identity component of the algebraic group Aut($\mathfrak{g}$).\*