We consider 2 different invertible matrices $A_{1}$ and $A_{2}$, and a submultiplicative norm $\|.\|$ such that $\|A_{1}\|< 1$ and $\|A_{2}\|< 1$. My goal is to find: $$ \sup\limits_{i_{1},i_{2},i_{3}i_{4}\in \mathbb N^{*}} \|A_{1}^{i_1}A_{2}^{i_2}A_{1}^{i_3}A_{2}^{i_4}\| $$
My intuition says that the supremum is achieved when $i_{1}=i_{2}=i_{3}=i_{4}=1$. In the case of commuting $A_{1}$ and $A_{2}$, it is easy to prove it, but when it not the case i don't know how to start, also i don't find a counter-example to disprove it. Does anyone can help?