Exercise 5, § 6

Let $g$ be a nilpotent Lie $\mathbf{Q}$-algebra which we give the Hausdorff group law, denoted by $(x, y) \mapsto x.y$.

(a) Show that, if $x \in g, y \in g$, then

(Use Exercise 2.)

(b) Let $\mathfrak{h}$ be a subset of $g$. Show that $\mathfrak{h}$ is a Lie subalgebra (resp. an ideal) of $g$ if and only if it is a saturated subgroup (resp. a saturated normal subgroup) of the group $g$. (Use the formulae of Exercise 4 (d) to pass from the group law to the Lie algebra law.)

(c) Let $\mathfrak{h}$ be a subgroup of the group $g$. Show that the $\mathbf{P}$-saturation of $\mathfrak{h}$ is $\mathbf{Q}.\mathfrak{h}$.

(d) Let $\mathfrak{h}$ be a Lie subalgebra of $g$. Show that the centralizer (resp. normalizer) of $\mathfrak{h}$ in the group $g$ is the set of $x \in g$ such that $(\mathrm{ad}\, x)(\mathfrak{h}) = 0$ (resp. $(\mathrm{ad}\, x)(\mathfrak{h}) \subset \mathfrak{h}$).

(e) Show that the lower central series of the Lie algebra $g$ coincides with that of the group $g$ and that the associated graded Lie algebra $\mathrm{gr}(g)$ is the same from the "group" point of view as from the "Lie algebra" point of view.