I find myself needing to cross two pairs of vectors, and cross that result (so a normal of normals) and check whether each of two of the original points are on different sides of the plane it defines:
$$ (A\times B)\times (C\times D) \cdot A \lt 0 \\ (A\times B)\times (C\times D) \cdot B \gt 0 $$
Since the ultimate result is a scalar, It'd be great if I could eliminate the cross products altogether (similar to how Lagrange's Formula does for the triple product).
Does anyone know of a similar identity for a scalar quintuple product?