I'm trying to understand Lemma 3.2 from p. 355 of the paper R. C. Carlson and K. R. Goodearl, Commutants of Ordinary Differential Operators, Journal of Differenial Equations 35 (1980), 339–365.
The authors define $B$ as a matrix coefficient with $C^\infty$ entries, $D$ is the ordinary derivative operator on $\bigoplus^k C^\infty(\mathscr J)$ for some open interval $\mathscr J$, and $L$ as the differential operator given below (from the previous page)
I can't see how this is true that any differential operator can be written in this way. I'm trying to see it in the scalar case, i.e. where $A,B$ are $1\times 1$ matrices but I can't even construct non-trivial examples that can be written in that way. Moreover, I don't even understand what the proof is doing.