In tackling a question in analysis, the following subset of $M_{n}(\mathbb{R}) \times M_{n}(\mathbb{R})$ (pairs of $n \times n$ real matrices) showed up:
$$\{(A, B)\in M_{n}(\mathbb{R}) \times M_{n}(\mathbb{R}) \mid (A^{k} - B^{k})(A^{l} - B^{l}) = 0 \text{ }\forall\text{ }k, l \geq 1 \} $$
Can this set be described in a different, perhaps simpler way? Is there a specific name for this set in linear algebra, and if so, can one point me to a reference?