Exercise 5, § 6

Let $g$ be a nilpotent $\mathbf{Q}$-Lie algebra, which is provided with the Hausdorff group law, denoted $(x, y) \mapsto x . y$. (a) Prove that, if $x \in g$, $y \in g$, one has

(Use Exercise 2.)

(b) Let $\mathfrak{h}$ be a subset of $g$. Prove that $\mathfrak{h}$ is a Lie subalgebra (resp. an ideal) of $g$ if and only if it is a saturated subgroup (resp. a saturated distinguished subgroup) of the group $g$. (Use the formulas of Exercise 4 d) to pass from the group law to that of the Lie algebra.) (c) Let $\mathfrak{h}$ be a subgroup of the group $g$. Prove that the $\mathbf{P}$-saturation of $\mathfrak{h}$ is $\mathbf{Q}.\mathfrak{h}$. (d) Let $\mathfrak{h}$ be a Lie subalgebra of $g$. Prove that the centralizer (resp. the normalizer) of $\mathfrak{h}$ in the group $g$ is the set of the $x \in g$ such that $(\mathrm{ad}\,x)(\mathfrak{h}) = 0$ (resp. such that $(\mathrm{ad}\,x)(\mathfrak{h}) \subset \mathfrak{h}$). (e) Prove that the descending central sequence 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 point of view of “group” and from the point of view of “Lie algebra”.