Corollary

Tag 0AEA, permanent. Cite it as [Bourbaki, Tag 0AEA].

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

Let $G$ be a finite-dimensional simply connected Lie group. There exists a finite-dimensional analytic linear representation of $G$ whose kernel is discrete.

There exist (Chapter I, § 7, Theorem 2) a finite-dimensional vector space $E$ and an injective morphism $h$ of $L(G)$ into the Lie algebra $\mathrm{End}(E)$ . By Theorem 1, there exists a morphism $\phi$ of $G$ into $\mathbf{GL}(E)$ such that $L(\phi) = h$ . Hence $\phi$ is an immersion and therefore its kernel is discrete.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Exercise 5, § 8
Proposition 8, § 4

Cites

Chapter I, § 7, Theorem 2
Theorem 1