Classification of degree 2 polynomial automorphisms of $\Bbb C^3$

For the family of degree at most $2$ polynomial self-maps of $\Bbb C^3$ with nowhere vanishing Jacobian determinant, we give the following classification: for any such map $f$, it is affinely conjugate to one of the following maps:

(i) An affine automorphism;

(ii) An elementary polynomial autormorphism
$$
E(x,y,z)=(P(y,z)+ax,Q(z)+by, cz+d),
$$
where $P$ and $Q$ are polynomials with $\max\{\deg(P),\deg(Q)\}=2$ and $abc\ne 0$.

(iii)
$$
\cases
H_1(x,y,z)=(P(x,z)+ay,Q(z)+x,cz+d)\\
H_2(x,y,z)=(P(y,z)+ax,Q(y)+bz,y)\\
H_3(x,y,z)=(P(x,z)+ay,Q(x)+z,x)\\
H_4(x,y,z)=(P(x,y)+az,Q(y)+x,y)\\
H_5(x,y,z)=(P(x,y)+az,Q(x)+by,x)
\endcases
$$
where $P$ and $Q$ are polynomials with $\max\{\deg(P),\deg(Q)\}=2$ and $abc\ne 0$.