How do we add two parallelograms in 4D geometric algebra?

Question

In 3D, we connect two of the sides and add the other two sides the usual vector way. But in 4D, say, the planes $e_1e_2$ and $e_3e_4$ do not intersect along any line. So now we cannot connect any lines to carry out the addition. So how do we add then?

Answer 1

What you've identified is what is known as a non-simple bivector which cannot be decomposed into the exterior product of two vectors. Non-simple bivectors exist in all dimensions greater than three.

Considering projective geometric algebra for the moment $\mathbf{P}(\mathbb{R}^*_{3, 0, 1})$, the product of two vectors which produces a bivector can be thought of as a line (a vector models a single reflection in 3D, manifested as a plane. The product of two vectors models two reflections, manifested as a line). A non-simple bivector in this view represents two skew lines. As such, they cannot be immediately exponentiated to generate a motor action. However, in this space, every non-simple bivector can, at the very least, be decomposed into the sum of a line and a line orthogonal to it. This decomposition then permits an exponential map as the action of the two lines now commute.

To see the derivation of the decomposition above, feel free to refer to the section $\S8.1$ in the SIGGRAPH PGA course notes linked here.

Answer 2

e1e2 + e3e4 is just that; there is no way to simplify it to represent a parallelogram.