Exercise 25, § 6

Lie Groups and Lie Algebras, III, 390

Bourbaki marks this one as harder than the rest.

Let G be a finite-dimensional real or complex Lie group. Let S ⊂ L(G). Suppose that L(G) is the Lie algebra generated by S and that S is stable under homotheties.

(a) Let H be the subgroup of G generated by exp S. Show that H is open in G. (Let A be the set of x ∈ L(G) such that exp(Kx) ⊂ H and B the vector subspace of L(G) generated by A. Show that [A, S] ⊂ A and then that [A, A] ⊂ B. Deduce that B is a subalgebra of L(G) and then use Proposition 3 of § 4.)

(b) Suppose further that

(x \in S \text{ and } y \in S) \Rightarrow ((\mathrm{Ad}\exp x)(y) \in S).

Then the vector subspace V of L(G) generated by S is equal to L(G). (Using (a), show that [L(G), V] ⊂ V.)

Solution

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