Differential of a Lie group homomorphism

Let $\Phi:G\to H$ be a homomorphism of Lie groups. Since $\Phi$ is a smooth map, it has a differential at each point; in particular, at the identity element $e_G\in G$ one gets a linear map

Using the identification of these tangent spaces with the Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, the map

is called the differential of the Lie group homomorphism.

A fundamental property is that $\phi$ respects brackets:

so $\phi$ is a Lie algebra homomorphism (the bracket on each side is the one induced from the Lie group structure). This construction is functorial: if $\Psi:H\to K$ is another Lie group homomorphism, then $\mathrm{d}(\Psi\circ\Phi)_{e_G}=\mathrm{d}\Psi_{e_H}\circ \mathrm{d}\Phi_{e_G}$.

Examples

1. Inclusion of a Lie subgroup. The inclusion $\iota:\mathrm{SO}(n)\hookrightarrow \mathrm{GL}(n,\mathbb{R})$ differentiates to the inclusion $\mathfrak{so}(n)\hookrightarrow \mathfrak{gl}(n,\mathbb{R})$.
2. Determinant. The map $\det:\mathrm{GL}(n,\mathbb{R})\to \mathbb{R}^\times$ is a Lie group homomorphism. Identifying $\mathrm{Lie}(\mathbb{R}^\times)\cong\mathbb{R}$, its differential at the identity sends a matrix $A$ to $\mathrm{tr}(A)$.
3. Adjoint action. The adjoint action $\mathrm{Ad}:G\to \mathrm{GL}(\mathfrak{g})$ is a Lie group homomorphism; differentiating it at the identity yields the adjoint Lie algebra representation $\mathrm{ad}:\mathfrak{g}\to \mathrm{End}(\mathfrak{g})$.