4-dim. generalization of $ab+ac+bc=0$ The equation $ab+ac+bc=0$ can be parameterized by $(a,b,c)=\lambda(-pq, p(p+q), q(p+q))$.
Is there a (similar) parameterization for $ab+ac+ad+bc+bd+cd=0$?
What about the 5-dimensional case?
Edit:
Proof of the parametrization in the 3-dim. case:
Write $x=a+b, y=a-b$.
Then $ab+ac+bc=0 <=> (x-y)(x+y) +2(x+y)c+2(x-y)c=0 <=> (x+2c)^2-y^2-4c^2=0$. Then use the pythogorean triple parametrization to get $x+2c=\lambda(m^2+n^2), y=\lambda(m^2-n^2), c=\lambda mn$.
Now $a=(x+y)/2= \lambda(m^2+n^2-2mn+m^2-n^2)/2=m(m-n), b=(x-y)/2=\lambda n(n-m), c=\lambda mn$. Now set $m=p$ and $n=q+p$ and the result follows (with $b$ and $c$ switched)
Edit2: yes, I am looking for a parameterization in the integers. Sorry, that I didn't make myself clear.

Solution of above inquiry regarding parametric form for equations (given below) for four and five variables is shown: 
$ab+ac+ad+bc+bd+cd=0$
$ab+ac+bc+bd+ad+cd+ea+eb+ec+ed=0$
For $ab+ac+ad+bc+bd+cd=0$
$(a,b,c,d)=((p^2+2p+1),(2p+2),-(2p+2),(4))$
$p=2$     we get $(a,b,c,d)=(9,6,-6,4)$
For $ab+ac+bc+bd+ad+cd+ea+eb+ec+ed=0$
$(a,b,c,d,e)=((3p+7),-3(p-3),(3p-5),-(3p+5),(3p^2-p+12))$
For $p=0$  we get:  $(a,b,c,d,e)=(7,9,-5,-5,12)$
