If $A \oplus_{\infty} B$ is isometric isomorphic to $C \oplus_{\infty}D,$ then $A \cong C$ and $B \cong D$ or $A \cong D$ and $B \cong C$?

Question

Suppose that $A,B,C$ and $D$ are Banach spaces. Denote $A\oplus_{\infty}B = \{ (a,b):\|(a,b)\|_{\infty} = \max\{ \|a\|,\|b\| \} \}.$ Let $A \cong B$ means that $A$ is isometric isomorphic to $B.$

Question: If $A \oplus_{\infty} B$ is isometric isomorphic to $C \oplus_{\infty}D,$ then is it true that $A \cong C$ and $B \cong D$ or $A \cong D$ and $B \cong C$?

I have a feeling that this should be true. But I have no idea how.

Answer 1

Hint: $$(A\oplus_{\infty} B)\oplus_{\infty} C \cong A\oplus_{\infty}( B\oplus_{\infty} C)$$