We give a projective construction for general non-singular algebraic plane cubics using a (2,2) correspondence between two line pencils. When the pencils are equipped with homographic involutions, the residual locus of their intersections---obtained by removing the baseline---is a cubic curve passing through the centers of both pencils. We also describe the tangency properties of the fixed lines of these involutions relative to the total quartic locus.
Hussein Khayou (Wed,) studied this question.