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