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. 



Existence of Cartan subalgebra
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.


