# Existence of Cartan subalgebra

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.

• Are you claiming that every element of a semisimple Lie algebra is a semisimple element? This is certainly false (consider $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \in \mathfrak{sl}_2$). Oct 28 '11 at 17:58

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
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 $x^{[p]}$).