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.