Consider a Hermitian matrix $B$ and a set of Hermitian matrices $\{A_i\}_{i=1}^n$. Now consider the nested commutator: $$ [A_n, [A_{n-1}, \dots [A_1,B]]. $$
I want to put an upper bound on the operator norm of nested commutator which depends on $||B||$ separately, so something like: $$||[A_n, [A_{n-1}, \dots [A_1,B]]|| \leq 2||B|| \ ||[A_n, [A_{n-1}, \dots [A_2,A_1]]||.$$
Is such an expression possible? This isn't obvious to me as the $B$ matrix is nested in the "center" of the commutator.