Reference to line parametrization Defining two lines in space, $\mathbb{R}^3$, as:

$l_1: \textbf{a}_1+\lambda_1\textbf{b}_1$

$l_2: \textbf{a}_2+\lambda_2\textbf{b}_2$
The line to line intersection condition is:
$\textbf{b}_1\cdot ((\textbf{a}_2-\textbf{a}_1)\times\textbf{b}_2)=0$
which can be rewritten as:
$\textbf{b}_1\cdot (\textbf{a}_2\times\textbf{b}_2)+\textbf{b}_2\cdot (\textbf{a}_1\times\textbf{b}_1)=0$
and 
$\begin{bmatrix} \textbf{b}_1\\\textbf{a}_1\times\textbf{b}_1\end{bmatrix}^T\begin{bmatrix} \textbf{a}_2\times\textbf{b}_2\\ \textbf{b}_2\end{bmatrix}=0$
This representation inspires new parametrization for a line $\textbf{a}+\lambda\textbf{b}$ defined as:
$L = \begin{bmatrix}  \textbf{b} \\ \textbf{a}\times\textbf{b}\end{bmatrix}$
(a given vector $L$ defines a line in space) 
Then the line-line intersection condition has a nice form:
$L_1^T\begin{bmatrix} 0&I\\I&0\end{bmatrix}L_2=0$
I remember this trick, but i don't remember the name of this parametrization.
Please help to find any reference...
 A: Up to order of coordinates, and perhaps some signs (depending on the convention you follow), this looks like Plücker coordinates.
I think about this as the line joining the point $\mathbf a$ with the point at infinity in direction $\mathbf b$. In homogeneous coordinates you could write both as the columns of a matrix:
$$\begin{pmatrix}
a_1 & b_1 \\
a_2 & b_2 \\
a_3 & b_3 \\
1 & 0
\end{pmatrix}$$
(I use the indices $1$ through $3$ to denote coordinate directions for the same line, not several distinct lines.) Now the Plücker coordinates of this are the values of all the $2\times2$ minors, namely
\begin{align*}
l_{12}&=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix} &
l_{13}&=\begin{vmatrix}a_1&b_1\\a_3&b_3\end{vmatrix} &
l_{23}&=\begin{vmatrix}a_2&b_2\\a_3&b_3\end{vmatrix} \\
l_{14}&=\begin{vmatrix}a_1&b_1\\1&0\end{vmatrix}=-b_1 &
l_{24}&=\begin{vmatrix}a_2&b_2\\1&0\end{vmatrix}=-b_2 &
l_{34}&=\begin{vmatrix}a_3&b_3\\1&0\end{vmatrix}=-b_3
\end{align*}
So you have
$$\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}
\times\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}=
\begin{pmatrix}\phantom+l_{23}\\-l_{13}\\\phantom+l_{12}\end{pmatrix}
\qquad
\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}=
\begin{pmatrix}-l_{14}\\-l_{24}\\-l_{34}\end{pmatrix}
$$
and can see that indeed, except for some sign changes and the order of terms, the two systems are equivalent.
By the way, to decide whether two lines $g$ and $l$ intersect, I'd use
\begin{align*}
g_{12}l_{34}-g_{13}l_{24}+g_{14}l_{23}+g_{23}l_{14}-g_{24}l_{13}+g_{34}l_{12}&=0\\
(+g_{23})(-l_{14})+(-g_{13})(-l_{24})+(+g_{12})(-l_{34})+(-g_{14})(+l_{23})+(-g_{24})(-l_{13})+(-g_{34})(+l_{12})&=0
\end{align*}
So this is indeed equivalent to the matrix notation you used.
I've seen various conventions for the order and signs of Plücker coordinate vectors, and almost all of them have benefits to justify their use. Therefore I see such conventions as only minor differences of essentially the same concept, and would confidently call your parametrization Plücker coordinates, even though it represents a convention unlike any I've seen before.
