Let $A$ be a $2 \times 2$ matrix with different (not necessarily real) eigenvalues and real coefficients. Prove that the matrices N such that the $4 \times 4$
$M = \begin{pmatrix}A & N \\ 0 & A \end{pmatrix}$ is diagonalizable over $\mathbb{C}$ form a 2-dimensional subspace of the space of $2 \times 2$ real matrices.
Call the block matrix in question $B$. Then $B$ diagonalisable over $\mathbb C$ if and only if $B-\frac12\operatorname{tr}(A)I_4$ is diagonalisable over $\mathbb C$. So, we may assume without loss of generality that $A$ is traceless. Let $p$ be the characteristic polynomial of this traceless $A$. Using the fact that $A$ has distinct complex eigenvalues, prove that $B$ is diagonalisable over $\mathbb C$ if and only if its minimal polynomial is equal to $p$. Hence show that in this case, $$p(B)=\pmatrix{0&AN+NA\\ 0&0}.$$ Consequently you need to show that when $A$ is traceless and it has distinct complex eigenvalues, the set of all real matrices $N$ such that $AN+NA=0$ is a two-dimensional subspace of $M_2(\mathbb R)$.