Corollary 1

(i) $\mathscr{D}^i\mathfrak{g}=\mathscr{D}^ie(\mathfrak{g})$ for $i\geq 1$, and $\mathscr{C}^i\mathfrak{g}=\mathscr{C}^ie(\mathfrak{g})$ for $i\geq 2$.

(ii) If $\mathfrak{g}$ is commutative (resp. nilpotent, resp. solvable), then $e(\mathfrak{g})$ is commutative (resp. nilpotent, resp. solvable).

Assertion (i) follows from Prop. 4 by induction on $i$ and (ii) follows from (i).