I'm trying understand this proof:
Obviously, $b_s\in \mathfrak a$, to see this just take $g_1=0$, but I couldn't prove why the others $b_i$ are in $\mathfrak a$. If we do as the author said we had $-b_sa_{d-1}+b_sb_{s-1}\in \mathfrak a$ which doesn't help to prove $b_{s-1}\in \mathfrak a$.
I'm sure it's just a silly thing, I need just a little help here in this elementary abstract algebra doubt.
Any help would be welcome.
Thanks in advance.