In Clifford Algebras and Spinors by Pertti Lounesto it is written the following about bivectors in four dimensions:
Now, I am wondering, how many different decompositions there are in the non-unique case. Is there any result about this? I am especially interested in the case where the four vectors $u_1, u_2, u_3, u_4$ such that $e_1 \wedge e_2 + e_3 \wedge e_4 = u_1 \wedge u_2 + u_3 \wedge u_4$ are all pairwise orthogonal and of the same length, i.e. $u_1, u_2, u_3, u_4$ is again an orthonormal base of $\mathbb{R}^4$.
I could find the decomposition $1/2(e_1 + e_4)\wedge(e_2 - e_3) + 1/2(e_1 - e_4)\wedge(e_2 + e_3)$. Are there really only these three unique (up to permutation and sign) decompositions?