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

Using the matrix-unit commutator and summing the adjoint indices givesEquivalently, in components this is $2n{X}^i{}_j{Y}^j{}_i-2X^i{}_i{Y}^j{}_j$. It vanishes on the scalar center, as expected because ${\mathfrak{gl}}_n$ is not semisimple.