I'm currently reading Hilgert, Neeb, Lie-Gruppen und Lie-Algebren, 1991, which is in some sense a predecessor to Hilgert, Neeb, Structure and Geometry of Lie Groups, 2012. As the book is in German I'll be here a bit more elaborate.
In preparation for the Baker-Campbell-Hausdorff formula the adjoint map or adjoint representation $Ad$ is defined. The authors consider the matrix Lie group $G = GL(n, \mathbb{K})$ and its associated Lie algebra $\mathfrak{g} = \mathfrak{gl}(n, \mathbb{K})$ and define $$ Ad(g) X = gXg^{-1} $$ with $g\in G$ and $X \in \mathfrak{g}$. One shows that $$ Ad: G \to GL(\mathfrak{g}) $$ where $GL(V)$ denotes the invertible linear maps on a vector space $V$. Proposition I.4.2 states that $Ad$ is a differentiable group homomorphism and a bit more.
In the proof the differentiability of $Ad$ in 1 is shown and than it is claimed that the differentiability in any other point can be derived from the fact that $$ Ad = \lambda_{Ad\ h} \circ Ad \circ \lambda_{h^{-1}} $$ where $\lambda_h$ is the left action on $G$; $\lambda_h g = hg$. Here, I reproduce the formula as shown in the book. But I assume that it should be $Ad(h)$ and is just a minor inconsistency in the notation.
Now, reading the formula from the right, I understand the first two parts, giving $$ g \mapsto h^{-1} g \mapsto h^{-1} g \bullet (h^{-1} g)^{-1} = h^{-1} g \bullet g^{-1} h $$ where the bullet should indicate that the expression is a map. Of course, the map $$ h^{-1}g \bullet g^{-1}h \mapsto h h^{-1} g \bullet g^{-1} h h^{-1} = g \bullet g^{-1} $$ does exactly what is needed for the result. But would that be the correct understanding of the definition of $\lambda_{Ad(h)}$? Or is it the consequence of a definition I don't see? To boil it down to my fundamental lack of understanding:
What would be the definition of $\lambda_{Ad\ h}$ on $GL(\mathfrak{g})$?
If there is a related question math.SE I have overlooked with an answer that also solves my problem a hint to that answer is of course sufficient. Otherwise, any help is appreciated.