The “form” of a triangle

Nevertheless, a 2 × 2 determinant would be less surprising* than a 4 × 4 determinant. What follows originates from work ([AC], see also [A3] and [C1]) done in collaboration with Alain Albouy on the symmetries of the n-body Problem and their reduction. That work generalizes Lagrange’s fundamental memoir [L] to more than three bodies. In it one understands that the above determinant is simply a means of computing a subtler determinant, namely that of an endomorphism of a two-dimensional vector space which possesses no privileged basis. (Think of the plane of equation x + y + z = 0 in IR. This plane is naturally equipped with the three lines of intersection with the coordinate planes x = 0, y = 0 and z = 0 but certainly not with a canonical basis.) Choices or tricks are necessary to compute in such a plane.