Consider the following space of operators on function of $n$-variables
$A= Span \{x_ix_j\ , x_i \frac{\partial}{\partial x_j} , \frac{\partial^2}{\partial x_i \partial x_j} , i,j=1,2,\cdots,n\}$. Prove that $A$ is closed under the commutator operation $[a,b]=ab-ba$.
I have tried to computed the bracket $[,]$ for the three combinations in the generators of span, but i actually dont know what is meant by closed under the operation, can someone compute one of them for me as an example, thank in advance.