The $6\times6$ 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 $ \operatorname{rank}[S,T]\leq2 \quad(S,T\in\mathcal V). $ We prove that $\mathcal V$, or its transpose, is conjugate to the algebra $ \left\{ \begin{pmatrix} A&B&C\\ 0&λI_2&D\\ 0&0&λI_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ λ\in\mathbb C \right\}. $ Consequently, the corresponding closed algebraic locus in $\operatorname{Gr}(17,M_6(\mathbb C))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\operatorname{Fl}(2,4;6)$. 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 $\mathcal A$.