In [a paper](http://link.springer.com/article/10.1023/A:1004157323726#page-1) 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.*



How was this Lie algebra found?
I would try the following: observe that $E$ can also be described as the subgroup of $\text{Aff}\_7$ under which the pullback of the polynomial $p:=z-x^2-y^2$ vanishes up to oder 3 along the zero set $z-x^2-y^2=0$. This translates into: $p$ is pulled back to an element in the ideal generated by $p$ up to terms of oder 3 or more (so formally an element of the ideal $(p)$ in the quotient ring of polynomials modulo ideal $(x^3,y^3,z^3)$). If we let $V$ denote the Lie algebra of $\text{Aff}\_7$ and $W$ the Lie Algebra of $E$, then in terms of Lie algebras the above means that $W$ consists of those derivations $v\in V$ which applied to $p$ give an element in $(p)$ mod $(x^3,y^3,z^3)$. Now the problem should reduce to a linear algebra problem by noting that $V$ has generators $z\partial\_z, x\partial\_x, y\partial\_x, z\partial\_x, x\partial\_y, y\partial\_y, z\partial\_y$ and the ideal $(p)$ mod $(x^3,y^3,z^3)$ has a vector basis $p,z\cdot p,x\cdot p,y\cdot p$. This might need some expanding. I haven't done these computations but checked that the basis given in your question satisfies $v\_1(p)=2xp+P\_3$, $v\_2(p)=2yp+P\_3$, $v\_3(p)=-2p$ and $v\_4(p)=0$.


