Non-abelian Cartan subalgebras

Question

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?

Answer 1

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.

Reference: Cartan subalgebras of simple nonclassical Lie p-algebras.

Answer 2

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.