Let $M_n$ denote the variety of $n \times n$ matrices over $\mathbb{C}$. Consider the algebraic subvariety of $M_n \times M_n \times \mathbb{C}^n \times \mathbb{C}^n$ defined by the equation $ab-ba + xy=0$, where $a$ and $b$ are $n\times n$ matrices and $x$ and $y$ are respectively column and row vectors so that their product is a rank one $n\times n$ matrix. I want to show that this variety isn’t irreducible, but I can’t find the proof even when $n=2$. Any help would be appreciated.
I know that if the commutator of 2 complex matrices has rank one, I can simultaneously triangulate them. Could this fact help me in the proof?
(The origin of the above equation is the study of representations of the Jordan quiver and the associated Hamiltonian reduction. Comment if you are interested in details.)