# Problem with Lee's proof on the derivative of adjoint representation

Background: suppose $$F:G\to H$$ is a homomorphism of Lie groups. Then, there is an induced homomorphism $$F_*:\mathfrak g\to \mathfrak h$$ defined by

$$Y_h:=(F_*X)_h=d(L_h\circ F_e)X_e \tag1\label{eq:1}$$

This definition is the one it makes sense to use because it makes $$X$$ and $$Y\ F$$-related.

Now, if $$X\in \mathfrak g,$$ we have $$X_e=\gamma'(0)$$ where $$\gamma(t)=\exp tX$$, which means that

$$(F_*X)_e=dF_e(X_e)=dF_e(\gamma'(0))=\frac{dF(\exp tX)}{dt}|_{t=0}\tag{2}\label{eq:2}$$

Now, apply \eqref{eq:1} and \eqref{eq:2} to the following data: fix $$g\in G$$ and set $$C_g:G\to G: x\mapsto gxg^{-1}$$ and $$\text{Ad}:G\to GL(\mathfrak g):g\mapsto (C_g)_*.$$ Then, $$\text{Ad}_*:\mathfrak g\to \mathfrak gl(\mathfrak g)$$ satisfies

$$(\text{Ad}_*X)_Y=d(L_Y\circ \text{Ad}_e)X_e \tag{3}\label{eq:3}$$

$$(\text{Ad}_*X)_e=\frac{d\text{Ad}(\exp tX)}{dt}|_{t=0}\tag{4}\label{eq:4}$$

My question is simply this: in the course of his proof, Lee has instead of \eqref{eq:4} the following equation

$$\text{Ad}_*X=\frac{d\text{Ad}(\exp tX)}{dt}|_{t=0}\tag{5}\label{eq:5}$$

which does not make sense to me. Now, if we take $$X$$ to be a vector in $$T_eG$$, then we can find a curve $$c$$ in $$G$$ such that $$c'(0)=X$$ and so we get

$$\text{Ad}_*X=\frac{d\text{Ad}(c(t))}{dt}|_{t=0}\tag{6}\label{eq:6}$$

and so if we take $$c=\gamma$$ where $$\gamma(t)=\exp tX$$, we get \eqref{eq:5}. But then, we are considering $$X$$ to be a left-invariant vector field again. Is this type of argument legitimate? If so, why? If not, why not?

I believe my confusion is that although the definition of the induced homomorphism $$F_*X$$ on a Lie algebra is obviously not the same as $$(F_*X)_e$$ and yet in defining the adjoint representation, Lee uses the former, in which case, I do not see clearly how $$(5)$$ obtains, whereas in another proof I have seen, one defines $$\text{ad}(X) = (d \text{Ad})_e(X)$$ (and then proves that $$\text{ad}(X)Y=[X,Y]$$) and then $$(5)$$ is just a consequence of the definiton. But of course, $$(d \text{Ad})_e(X)\neq \text{Ad}_*X$$.

There's an implicit identification going on here. For any finite-dimensional vector space $$V$$, let $$GL(V)$$ denote the Lie group of invertible linear transformations of $$V$$. There's a canonical Lie algebra isomorphism between the Lie algebra $$\operatorname{Lie}(GL(V))$$ of left-invariant vector fields on $$GL(V)$$ and the Lie algebra $$\mathfrak g\mathfrak l(V)$$ of endomorphisms of $$V$$ (under commutator bracket). (See Corollary 8.42 in my Introduction to Smooth Manifolds, 2nd ed.)
The equation you labeled as (5) above is using this identification for the Lie algebra of $$GL(\mathfrak g)$$. The right hand side is the initial velocity of a smooth curve in $$GL(\mathfrak g)$$. Considering the latter as an open subset of the vector space $$\mathfrak g\mathfrak l(\mathfrak g)$$, the velocity vector can be regarded unambiguously as an element of the vector space $$\mathfrak g\mathfrak l(\mathfrak g)$$ itself.