I am trying to prove the following (or some variant of the following):
$$|| (A \otimes B \otimes C) ~v|| \leq ||B|| \cdot || (A \otimes \mathbb{1} \otimes C) ~v|| $$,
where $A$, $B$ and $C$ are real valued matrices and $\mathbb{1}$ is an identity operator on the same space that $B$ belongs to (you can just assume them to be $2x2$ matrices if that's convenient). $|| \cdot ||$ is the Frobenius norm. $v$ is a vector of appropriate dimensions with $||v|| = 1$.
I know that
$$|| A \otimes B \otimes C|| \leq ||A||~ ||B|| ~||C|| $$ but that's not sufficient to reach the result.