In some lecture notes of mine we define a Cartan subalgebra $\mathfrak h$ for semisimple $\mathfrak g$ as an abelian subalgebra of $\mathfrak g$ containing ad-diagonizable elements which are maximal.
It then says that for more general Lie algebras $\mathfrak g$, a Cartan subalgebra is defined as a self-normalising nilpotent subalgebra. It then goes on to say that this is automatically maximal among nilpotent subalgebras.
My question is how would one show this (that it's maximal)? It says automatically but I can't see how it follows. Thanks in advance.