We say that a matrix $J \in \mathbb{R}^{n \times n}$ is nilpotent if $J^n = 0$. This is equivalent to the statement that $\forall x \in \mathbb{R}^{n} \quad \exists k \in \mathbb{Z}^+$ such that $J^kx = 0$. What I would like to do is to extend this notion to pairs of matrices in the following way.
$J_1, J_2 \in \mathbb{R}^{n \times n}$ is a nilpotent pair if $$\forall x \in \mathbb{R}^{n} \quad \exists N \in \mathbb{Z}^+, \{i_1,k_1,\ldots,i_N,k_N\} \in {\mathbb{Z}^+}^{2N}$$ such that $$J_1^{i_N}J_2^{k_N}\ldots J_1^{i_1}J_2^{k_1}x = 0$$
I am not sure if this definition makes sense. What I would like to ask is how one would attempt to characterize such a notion, i.e. what kind of properties the matrices $J_1,J_2$ should satisfy to be called a nilpotent pair. My apologies if this trivially follows from some existing result.
There is a simple characterization indeed, very similar to what exists for just one nilpotent matrix.
A pair $(J_1,J_2)$ is jointly nilpotent iff one of the products $J_1^{i_N}J_2^{k_N}\ldots J_1^{i_1}J_2^{k_1}$ is the zero matrix.
Because if all those products are non-zero, then all their kernels are strict subpaces of ${\mathbb R}^n$. So the union of all those kernels (which is a countable union) cannot cover the whole of ${\mathbb R}^n$ (for example it is nowhere dense by the Baire category theorem).