Why is that an automorphism that preserves $B$ and $H$ an automorphism of $\Phi$ that leaves $\Delta$ invariant? Let $L$ be a semisimple finite dimensional Lie algebra, $H$ its CSA and $\Phi$ its root system with base $\Delta$ and $B = B(\Delta) = H\bigoplus_{\alpha \succ 0}L_\alpha$. If we have an automorphism of $L$ that keeps $B$ and $H$ invariant, why is it an automorphism of $\Phi$? (Why does that keep $\Phi$ invariant?) Further, why does it keep $\Delta$ invariant?

$\Phi$ is determined by the pair $(L,H)$, so if the automorphism preserves both $L$ and $H$ it must preserve $\Phi$.
Again, $\Delta$ is determined by the set of all positive roots. But from borel you can reconstruct positive roots (they are eigenvalues of $ad$ on $B$). 
