Existence of Cartan subalgebra

Question

I am reading Helgason's book. In Chapter 3 he proved the existence of Cartan subalgebra for a semisimple Lie algebra $\mathfrak g$ (definition: a Cartan subalgebra is a maximal abelian subalgebra all whose element $H$ satisfies $\text{ad}_H$ is semisimple).

It seems to me the proof is quick: if $H\in {\mathfrak g}$, then $\text{ad}_H$ is automatically semisimple because $K(\text{ad}_H X, Y)+K(X, \text{ad}_H Y)=0$, where the Killing form $K$ is nondegenerate (since $\mathfrak g$ is semisimple) - thus any $\text{ad}_H$ invariant subspace has an invariant complementary subspace. So any maximal abelian subalgebra in a semisimple Lie algebra is a Cartan subalgebra.

My question is, is the above argument valid? I am confused since Helgason spent more than 3 pages proving the existence of Cartan subalgebra in the semisimple case - of course his proof contains a lot of information.

Answer 1

This argument does not work because the Killing form is not generally positive definite, so the orthogonal of a subspace wrt the Killing form is not necessary a complement of the subspace.

Answer 2

Let me just add some remarks. In general, if $L$ is a finite dimensional Lie algebra over an arbitrary field $F$ then a subalgebra $H$ of $L$ is called a Cartan subalgebra if $H$ is nilpotent and self-normalising in $L$. If $L$ is semisimple and $F$ has characteristic zero (as in the case asked by the OP) then the Cartan subalgebras of $L$ are precisely the maximal tori of $L$. (A torus of $L$ is an abelian subalgebra consisting of semisimple elements). Note that the existence of a Cartan subalgebra is always assured whenever the ground field has more than $\dim_F L$ elements. In particular, finite dimensional Lie algebras over infinite field always have Cartan subalgebras. Moreover, the Cartan subalgebras coincides with the minimal Engel subalgebras of $L$. (A subalgebra of $L$ is called an Engel subalgebra if it is the null Fitting component of $L$ with respect to $ad x$ for some $x\in L$.) See the paper

R.E. Barnes: On Cartan Subalgebras of Lie Algebras, Math. Z. 101 (1967), 350-355.

On the other hand, the existence of Cartan subalgebras of Lie algebras defined over small fields still remains an OPEN problem.

It is also worth to mention that solvable Lie algebras always have Cartan subalgebras.

Finally, if $L$ is a finite dimensional restricted Lie algebra over a field of characteristic $p>0$, then $H$ is a Cartan subalgebra of $L$ if and only if is the centralizer of a maximal torus of $L$. (Here a torus is an abelian subalgebra consisting of semisimple elements; an element $x$ of $L$ is semisimple it $x$ is in the restricted subalgebra generated by $x^{[p]}$).