Let $D$ be a division ring. Let $V$ be a finite dimensional module over $D$, let $I \subseteq\operatorname{End}_D(V)$ be a $D$-submodule on both sides (I mean a subgroup closed by both on left and right multiplication of scalar matrices), closed under multiplication, and consisting of nilpotent operators.
Question: May iI always find a basis of $V$ where all the elements of $I$ are in triangular form?