The $6\times 6$ equality case of matrix spaces with rank-two commutators

Let $\mathcal{V}\subseteq M_6(\mathbb{C})$ be a $17$-dimensional linear subspace such that $\mathrm{rank}[S,T]\le 2(S,T\in \mathcal{V})\mathrm{.}$ We prove that $\mathcal{V}$, or its transpose, is conjugate to the algebra $\{\left(\begin{array}{ccc}A& B& C\\ 0& \lambda I_2& D\\ 0& 0& \lambda I_2\end{array}\right):A,B,C,D\in M_2(\mathbb{C}),\lambda \in \mathbb{C}\}\mathrm{.}$ Consequently, the corresponding closed algebraic locus in $\mathrm{Gr}(17,M_6(\mathbb{C}))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\mathrm{Fl}(2,4;6)$. We also prove that the Zariski tangent space at $\mathcal{A}$ of the corresponding closed algebraic locus is equal to the tangent space to the conjugacy orbit of $\mathcal{A}$.