I'm trying to prove the following statement:
Given a Lie group $G$ with Lie algebra $\mathfrak{g}$, if $\psi: \mathfrak{g} \rightarrow \mathfrak{X}(\mathfrak{g})$ is the infinitesimal action determined by the adjoint action then $$ \psi \left( \left[ X , Y \right] \right) = - \left[ \psi(X) , \psi(Y) \right]\,. $$
Initially I assumed that the adjoint action was the usual $\mathrm{Ad}: G \rightarrow \mathrm{Aut}(\mathfrak{g})$ which would imply that $\psi = \mathrm{ad}$, however in this case, since we have that $\mathrm{ad}(X) (Y) = \left[X,Y \right]$ and then through the Jacobi identity one obtains
$$ \mathrm{ad} \left( \left[ X,Y \right] \right) (Z) = \left[ \mathrm{ad}(X) ,\mathrm{ad}(Y) \right] (Z) $$
which does not agree with what is asked. I guess I'm probably looking at the wrong thing but I can't understand what could also be meant by the adjoint action in that case.
Any pointers on what is $\psi$ (if indeed it is something different than what I interpreted it to be) or otherwise on what I might be doing wrong, would be greatly appreciated.