Constructing pairwise commuting nilpotent matrices

Question

How can I construct $K$ mutually commuting nilpotent matrices $A_i$ with nilpotent index 3? In other words, I need a set of matrices $A_i$ with following properties:

1. $A_i^3=0$ for $1\leq i \leq K$
2. $A_i A_j = A_j A_i$
3. $A_{i_1}^{n_1} A_{i_2}^{n_2} \cdots \neq 0 $ if $i_1 \neq i_2 \neq i_3 \cdots $ and $n_1 \leq 2, n_2 \leq 2, \cdots $

Answer 1

$\def\CC{\mathbb{C}}$ Consider the ring $A=\CC[X_1,X_3\dots,X_n]/(X_1^3,X_2^3,\dots,X_k^3)$ and for each $i\in\{1,\dots,k\}$ let $m_i:A\to A$ be the map given by multiplication by the class of $X_i$. It is clear that the linear maps $m_1,\dots,m_k$ satisfy your conditions, so you can take their matrices with respect to any basis of $A$.