I am self studying Humphrey's Lie algebra text and I am trying to understand the proof that the centralizer the maximal toral subalgebra is in fact equal to the subalgebra.
SO by definition, all elements of the toral subalgebra is semisimple, but then during part of the proof, we even prove that the centralizer is nilpotent:
But this is where I got confused, because since the centralizer contains $H$ and elements of $H$ are by definition semisimple, this would imply that every elements in $H$ is both semisimple and nilpotent. But the only elements where can be both semisimple and nilpotent is $0$. So this implies that $H=0$ which makes no sense. So what is going on?