In a paper the author lists, without justification, generators for a Lie algebra. *I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.*

Consider the three-dimensional affine space with a chosen coordinate system $(x, y, z)$ and the seven-dimensional group, $\text{Aff}_7$ of orientation-preserving linear transformations of the space that map the half-space $z \ge 0$ into itself.

The space of all $k$-jets of surfaces tangent to the plane $z = 0$ at the point $(0, 0, 0)$ is called the small space of $k$-jets and is denoted by $J_k.$

Consider the fibration $\pi : J_3 \twoheadrightarrow J_2$ where, for homogeneous polynomials $P_k$ of degree $k$, we have $\pi(P_2+P_3)=P_2$. Let $e$ be the fibre $\pi^{-1}(x^2+y^2) = x^2 + y^2 + P_3$. Let $E$ be the maximum subgroup of $\text{Aff}_7$ under which the fiber $e$ is invariant. The author goes on to say that a basis for the Lie algebra of $E$ has the basis:
$$\begin{eqnarray*}
v_1 &=& z\frac{\partial}{\partial x} \\ \\
v_2 &=& z\frac{\partial}{\partial y} \\ \\
v_3 &=& -\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+2z\frac{\partial}{\partial z}\right) \\ \\
v_4 &=& x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}
\end{eqnarray*}$$
As I mentioned above: *I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.*