As usually I'm trying to understand things. This time is the following Lemma 1.21. I report also some other stuff to help comprehension.
Anyway all the stuff is taken from Casolo
Question
I understand pretty much everything but I cannot see why in second to last line (of lemma 1.21) he says that $[B^{\langle x_q\rangle},\, \langle x_d^{q_d}\rangle]=[B,\, \langle x_d^{q_d}\rangle]$ (I think there is an misprint here). I think he applies the induction base to $[B^{\langle x_d\rangle},\, \langle x_d^{q_d}\rangle]$ since we need a subgroup, like $B^{\langle x_d\rangle}$, which is invariant by the action of $x_d$, but then I cannot understand the equality. Any suggestion?
Definition
By recurrence we define $[x,_0\, y]=x$; $[x, _{i+1}\, y]=[[x, _i\, y],\, y]$