$X = \sum_i a_i \frac {\partial}{\partial x_i}$, $Y = \sum_j a_j \frac {\partial}{\partial x_j}$. What is (Xg)Y? $a_i$, $a_j$, $g$ are differentiable functions. 
I don't know how to calculate $(Xg)Y$. My guess is using the chain rule but I'm not sure.
$(Xg)Y = (\sum_i a_i \frac {\partial g}{\partial x_i}) Y$ is my work so far.

To me, it looks like $Xg$ is just a function, so you can combine it with the coefficients in the definition of $Y$. 
Your notation doesn't distinguish $a_i$ from $a_j$ (the index is a dummy variable, so I will use $a_i$ for $X$ and $b_i$ for $Y$. Then 
$$(Xg)Y = \sum_j \left(\sum_i b_j a_i\frac{\partial g}{\partial x_i} \right)\frac{\partial }{\partial x_j}$$
unless I'm missing something here.
