The more general fact is that, if $N \subset \mathrm{Mat}_{n \times n}(k)$ is a subrng (ring without identity) whose every element is nilpotent, then we can choose a basis to make every element of $N$ simultaneously strictly upper triangular. This is a useful lemma which, as far as I know, doesn't have a standard name.

Note that, if $N_1$ and $N_2$ are commuting nilpotent matrices then any polynomial without constant term in $N_1$ and $N_2$ is clearly nilpotent, so this applies to the rng generated by such an $N_1$ and $N_2$.

**Proof:** Let $R$ be the $k$-algebra (with identity) generated by $N$, so $R = k \mathrm{Id} \oplus N$ and $N$ is the radical of $R$. Let $V$ be the vector space $k^n$, which we think of as an $R$-module. By the non-commutative Nakayama Lemma, also known as the Jacobson–Azumaya theorem, $NV \subsetneq V$. So $V \supsetneq NV \supsetneq N^2 V \supsetneq \cdots$ until we reach a power for which $N^r V=0$. Then choosing a basis of $V$ to respect the filtration by $N^r V$, we find that $N$ acts strictly upper-triangularly.