Inverse conjugation and adjoint antirepresentations ($F(gh)=F(h)F(g),F(g)={\mathrm{Ad}}_{g^{-1}}$)

With Adjoint representation convention ${\mathrm{ad}}_X(Y)=[X,Y]$, the matrix exponential identity is $e^{-X}Y{e}^X=e^{-{\mathrm{ad}}_X}Y$. Inverse conjugation reverses product order, so it is a right action or antirepresentation, whereas ${\mathrm{Ad}}_g(Y)=gY{g}^{-1}$ is an ordinary left group representation. The first-order terms $Y\mp [X,Y]$ detect a sign mismatch immediately. Negating every adjoint generator without reversing the Lie bracket does not give a new Lie-algebra homomorphism.