Exercise 8, § 9

Let G be a finite-dimensional connected real Lie group, g its Lie algebra, n the largest nilpotent ideal of g and h = [g, g] + n.

(a) h is a characteristic ideal of g; the radical of h is n; for all x ∈ h, Tr ad_h x = 0; for every Levi section l of g, h = l + n.

(b) Suppose that G is simply connected. Let Z be its centre, H the Lie subgroup of G with Lie algebra h and φ the canonical morphism of G onto G/H. Then φ(Z) is discrete in G/H. (The group G is the semi-direct product of a semi-simple Lie group S and its radical R. Let z ∈ Z. Then z = y^{-1}x, where x ∈ R and y belongs to the centre of S. There exists an integer p such that the eigenvalues of Ad_g y, and hence also of Ad_g x, are p-th roots of unity (Exercise 7). Deduce that, if N is the Lie subgroup of G with Lie algebra n, there exists a neighbourhood U of e in R satisfying U = UN = NU,

(c) We no longer assume that G is simply connected. Let H be the integral subgroup of G with Lie algebra h. Show that H is a unimodular normal Lie subgroup of G with nilpotent radical, such that G/H is commutative. (Use (a) and (b).)

(d) Deduce that, if (G, G) is dense in G, the radical of G is nilpotent.