In the book Brian C Hall, Lie Groups, Lie Algebras, and Representations, a complex matrix Lie group is a matrix Lie group $G \subset \mathrm{GL}_n(\mathbb{C})$ whose Lie algebra $\mathfrak{g} \subset \mathfrak{gl}_n(\mathbb{C})$ is a subspace of $\mathfrak{gl}_n(\mathbb{C})$ over $\mathbb{C}$, not only over $\mathbb{R}$.
Question: we know that a Lie group homomorphism $\phi : G \to H$ gives a Lie algebra homomorphism $\bar{\phi} : \mathfrak{g} \to \mathfrak{h}$ over $\mathbb{R}$. Is it true that $\bar{\phi}$ is linear over $\mathbb{C}$ if $G, H$ are complex matrix Lie groups?
If it is not true in general, then for a complex representation $\rho : G \to \mathrm{GL}_n(\mathbb{C})$ of a complex matrix Lie group $G$, $\bar{\rho} : \mathfrak{g} \to \mathfrak{gl}_n(\mathbb{C})$ is not a Lie algebra representation over $\mathbb{C}$. If we want it over $\mathbb{C}$, then we need to introduce another imaginary unit $j$ to construct $\mathfrak{g}_{\mathbb{C}}$ and $\bar{\rho} : \mathfrak{g}_{\mathbb{C}} \to \mathfrak{gl}_n(\mathbb{C})$. However, I don't think such a discussion is not in the book so I am confused (see Propositions 3.38 and 4.6).