# Non-abelian Cartan subalgebras

It is well-known that a Cartan subalgebra of a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0 is abelian.

I guess this is not true in general. So, I would like to find a finite field $\mathbb{F}_q$ and a Lie algebra $L$ over $\mathbb{F}_q$ such that $L$ has a non-abelian Cartan subalgebra.

Does anyone have any idea?

• @DietrichBurde Thank you very much. – NongAm Aug 18 '18 at 11:34

## 2 Answers

If we consider finite-dimensional semisimple Lie algebras, as you do, there are such Lie algebras over $\Bbb{F}_q$, which have non-abelian Cartan subalgebras. More precisely, there are such Lie algebras of Cartan type $K_n$ and $H_n$ having $n+1$ and $2n+1$ conjugacy classes of Cartan subalgebras, respectively.

Take any nilpotent Lie algebra $\mathfrak n$ (over any field) which is not abelian. Then $\mathfrak n$ is a Cartan subalgebra of $\mathfrak n$ which is not abelian.