There is a problem given in a representation theory textbook:
Prove that for any finite-dimensional complex vector space $V$ there are no $X, Y \in \operatorname{End}V$ such that $[X, Y] = \mathrm{id}$.
I tried looking at $\mathbb{C}[X, Y]$ and the ideal of $\operatorname{End}V$ generated by $XY - YX$, but so far to no avail. I could use a hint.