Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 2 / b

Conjugation differentiates to the Adjoint representationBecause conjugations compose, ${\mathrm{Ad}}_{gh}={\mathrm{Ad}}_g{\mathrm{Ad}}_h$ and ${\mathrm{Ad}}_e=I$, so this is a group representation.

Differentiating ${\mathrm{Ad}}_{\exp (tX)}Y$ at zero gives the Adjoint representation of a Lie algebraThe Jacobi identity implies $[{\mathrm{ad}}_X,{\mathrm{ad}}_Y]={\mathrm{ad}}_{[X,Y]}$, proving that it is a Lie-algebra representation.