I need to know which norms can be consistently decomposed in the sense that if I take a vector $ v = (a,b,c)$ and build a vector $ v’ = (\|(a,b)\|,c)$ we have $ \|v\| = \|v’\|$ .

More precisely, let $ v = \sum_{i=1}^n v_ie_i$ be a finite-dimensional vector, and $ \{P_j\}_{j=1}^k$ a partition of the index set $ \{i\}_{i=1}^n$ into $ k$ subsets. Then $ v = \sum_{j=1}^k \sum_{i \in P_j} v_ie_i$ . The question is for which norms is it true that $ $ \|v\| = \left\|\sum_{j=1}^k \Bigg\|\sum_{i \in P_j} v_ie_i\Bigg\|e_j\right\| $ $ for all vectors.

It is easy to see that this is true for every $ p$ -norm, and every other norm that I’ve tried failed to have this property, so it would be natural to conjecture that $ p$ -norms are the only consistently decomposable ones. Just finding a counterexample to this conjecture would be very useful.