Let $A, B \in \mathbb{R}^{d \times d}$ denote two symmetric positive definite matrices. I am interested in solutions $V_r \in \mathbb{R}^{d \times r}, 1 < r < d$ to the system of quartic polynomials $$ V_r^\top [A, B] V_r = [V_r^\top A V_r, V_r^\top B V_r] $$ and $$ V_r^\top V_r = I_r, $$ where $[A,B] = AB-BA$ denotes the matrix commutator. In other words, I wish to find orthonormal matrices $V_r$ whose quadratic form distributes over the commutator of $A$ and $B$.
By inspection we know this system is satisfied if $V_r$ corresponds to any $r$ eigenvectors of $A$, or if $V_r$ corresponds to any $r$ eigenvectors of $B$. However, you cannot combine subsets of eigenvectors of $A$ with $B$, except in the special case where they commute.
My conjecture is that these are the only solutions. Since my constraints are ``(real) algebraic varieties''---they are at most quartic degree polynomials in $d$ dimensions, and there are $r(r-1)$ such constraints in total---I suspect there must be tools which allow us to count the number of points where all of these curves intersect. In particular, if we can show that the number of intersections are less than or equal to $2 {d \choose r}$, then my conjecture follows.
I would appreciate if anybody could provide references to such theorems, or perhaps disprove my original conjecture!
