The <Adjoint representation of a Lie algebra> is
$$
\operatorname{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g),
\qquad \operatorname{ad}_X(Y)=[X,Y].
$$
It is linear. The <Jacobi identity> gives
$$
[\operatorname{ad}_X,\operatorname{ad}_Y]Z
=[X,[Y,Z]]-[Y,[X,Z]]=[[X,Y],Z]
=\operatorname{ad}_{[X,Y]}Z,
$$
so it is a <Lie algebra representation>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-302/3/a/solution