Lie group of differential operators

Question

I have the following three partial differential operators

$$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ $$C=y((1-z)\frac{\partial}{\partial z}-(a+b-c))$$

I found that the commentator relation of these operators are

$[A,B]=-B$ , $[A,C]=C $ and $$[B,C]=-z\frac{\partial}{\partial z}+(a+b-c)$$

Now to generate Lie group of these operators $[B,C]$ must be written as a linear combination of these operators

How could I write $[B,C]$ as a linear combination of these operators ?

If I add new operator $$D=z \frac{\partial }{\partial z}$$ Is the problem could be solved ??

Please , help me

Answer 1

"Adding $[B,C]$ then It should be compute it's relation with the other operators $A,B,C$" - No. Its relation to the other operators is $[B,C]=BC−CB$. It may turn out that $[B,C]$ is a linear combination of A,B,C, but there is no requirement that this has to be. My earlier comment was in response to your last question. There you suggest adding a different operator D to A,B,C. But why would you choose that particular operator instead of defining $D=[B,C]$ to be the new add. Then your solution is obvious.