Please follow the book here
Section: $16.4$ Proof of the theorem:
(1) When $B\cap B'\neq 0$
case (ii) $B\cap B'$ has no nonzero nilpotent elements. Let $T=B\cap B'$. Then the author shows that we may assume $T\subset H$.
My question: $B\cap B'\subset B$ and has no nonzero nilpotent elements in it and $B$ is standard with respect to CSA, $H$. And we know that a BSA contains $x_s$ and $x_n$, the semisimple and the nilpotent part of the elements $x$ in it. So $T\cap N=0$. Hence $T\subset H$.
Is there anything I've missed? I'm not getting why did he prove that "we may assume" ?