Are there a set of different $n$ matrices that commute that
1) after $n$ multiplications of these matrices - that is for example, $A \times B \times C \times ...A_n$, where $\times$ represents matrix multiplication, if the multiplied result was a product of some different matrices and two or more equal matrices, the result of multiplication is triangular. Otherwise, the result is not triangular.
Does this set exist for all numbers for $n$ bigger than 4?