Differentiate $\operatorname{Ad}_{e^{tX}}Y=e^{tX}Ye^{-tX}$ at zero. This gives the <Adjoint representation of a Lie algebra>
$$
\operatorname{ad}_X(Y)=[X,Y]=XY-YX.
$$
The <Jacobi identity> is exactly
$$
[\operatorname{ad}_X,\operatorname{ad}_Y]
=\operatorname{ad}_{[X,Y]},
$$
so this is a Lie-algebra representation on $\mathfrak o(3)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-302/1/e/solution