Lemma. The differential preserves the Lie bracket [liu2016lie, Section 2.1, p. 9] [fcap-001S]
✍️source

In the setting of Definition [fcap-001R], the differential is a Lie-algebra homomorphism: \[\operatorname {Lie}(\phi )([X,Y]) =[\operatorname {Lie}(\phi )(X), \operatorname {Lie}(\phi )(Y)].\] Indeed, the left-invariant fields \(\widetilde X\) and \(\widetilde Y\) are \(\phi \)-related to the left-invariant fields determined by their images. Brackets of related vector fields are again related. Evaluating that relation at \(e_G\) gives the displayed identity.

This argument is local at the identity. In particular, bracket preservation does not require \(G\) to be connected or simply connected.

Lemma. The differential preserves the Lie bracket [liu2016lie, Section 2.1, p. 9] [fcap-001S]
✍️source

Reference. Notes for lie groups and representations [liu2016lie]

Differentiating smooth group homomorphisms [fcap-001Q]

Definition. The differential of a smooth homomorphism [liu2016lie, Section 2.1, p. 9] [fcap-001R]

Lemma. The differential preserves the Lie bracket [liu2016lie, Section 2.1, p. 9] [fcap-001S]
✍️source

Theorem. The Lie functor [liu2016lie, Section 2.2, p. 10] [fcap-001T]

Theorem. Naturality of the exponential [liu2016lie, Section 2.4, pp. 12--13] [fcap-001U]

Proof.

Example. Differentiating a matrix representation [liu2016lie, Sections 2.1 and 2.4, pp. 9, 12] [fcap-001V]

Remark. Why explicit smoothness changes the roadmap [liu2016lie, Sections 2.1--2.4 and 2.8, pp. 9--13, 19--20]; [isaev2018theory, Section 3.1.2, pp. 106--107] [fcap-001W]

Theorem. The Lie functor [liu2016lie, Section 2.2, p. 10] [fcap-001T]

Definition. The differential of a smooth homomorphism [liu2016lie, Section 2.1, p. 9] [fcap-001R]