Nilpotent degree $2$ 'families' of $4\times 4$ matrices

Question

$\def\b{\begin{bmatrix}}\def\e{\end{bmatrix}}$

Are $\b0&0&a&b\\0&0&c&d\\0&0&0&0\\0&0&0&0\e$ and it's transpose, $a,b,c,d\in \Bbb C$ the only nilpotent degree $2$ 'families' of matrices of size $4\times 4$?

I believe they are, but I wanted to verify.

Answer 1

There are more forms. You will do well in considering Jordan blocks and in comparing the minimal polynomial to the characteristic polynomial.

One such example:

$$\begin{bmatrix}0&0&a&0\\0&0&0&0\\0&0&0&0\\0&b&0&0\end{bmatrix}=0_{4\times 4}$$