Any traceless $n\times n$ matrix with coefficients in a field of caracteristic $0$ is a commutator (or Lie bracket) of two matrices. What happens when the field has positive caracteristic?
When trying to reproduce the proof I have in the caracteristic $0$ case for the positive caractersitic case, I run into two problems:
- multiples of the identity may have trace $=0$.
- a matrix may have a spectrum equal to the whole field.
Are all traceless matrices commutators? If not, for which $n\in\mathbb{N}\setminus\lbrace 0 \rbrace$ and fields $k$ does it still hold? EDIT given a traceless matrix $M$, I want to know wether there are two matrices $A,B$ with $M=AB-BA$.