The procedure computes a line in the ℙ11 given by the polynomial ring Sa. This lines contains the given point and is completely contained in the variety of the 4 quadratic relations in ℙ11. If an second ideal J is given as an argument, the line will intersect the vanishing locus V(J), if possible. The line is represented by a 12×2 matrix whose first column parametrize the given first point and whose second column represents either a general point on the line or a point in the variety V(J). The row matrix "subsLine" contains all variables of the construction with the a-variables replaced by the assignment obtained from the chosen line.