Although this is not an answer to the original question, readers might be interested to know of a cute open problem in the context of commutative subalgebras of $M_n(\mathbb{C})$. Let $A$ be such an algebra; we focus on the minimal number of generators of $A$ as a $\mathbb{C}$ algebra vis-a-vis the dimension of $A$ as a $\mathbb{C}$ space. If $A$ is generated by just one element, then by Cayley-Hamilton, the dimension of $A$ as a $\mathbb{C}$ space is bounded above by $n$. It is an old theorem of Gerstenhaber as also Motzkin and Taussky-Todd that if $A$ can be generated by two elements, then too, the dimension of $A$ is bounded above by $n$. OTOH, if $A$ needs at least four generators, there are examples to show that the dimension of $A$ can be greater than $n$. So here is the open question: are there commutative subalgebras $A$ that can be generated by three elements for which the dimension of $A$ as a $\mathbb{C}$ space is greater than $n$?

The work on this question has led naturally to study the variety of commuting triples of matrices (a subvariety of affine 3n^2 space over $\mathbb{C}$). This variety has been proved to be reducible (by Guralnick) for $n\ge 32$ (since sharpened to $n\ge 29$ by others); one set of enquiries focuses on determining irreducibility of this variety for smaller $n$. I believe it is known now that for $n\le 7$, this variety is irreducible. This translates to the result that for $n\le 7$, such an $A$ as above must indeed have dimension bounded by $n$.

Given the reducibility for general $n$ of this variety, another set of enquiries focuses on studying specific subvarieties of this commuting triples variety. For instance, studying the varieties of commuting triples $(M_1, M_2, M_3)$ where $M_3$ is *fixed* to be a matrix in which each eigenvalue appears in at most two blocks (aka "$2$-regular") leads naturally to generalized tangent varieties over determinantal varieties. On the other hand, fixing $M_3$ to be the nilpotent Jordan block of size $k$ repeated $m$ times along the diagonal (so $n = mk$) leads to generalized tangent varieties over the variety of commuting *pairs* of $m\times m$ matrices.

Interesting stuff, this!