We are looking for a map from Ck to (Vk→Vk) which includes the modeling of orientations by Clifford numbers. We have conjectured that the map is the quadratic form corresponding to some bilinear map from Ck to (Vk→Vk). (a linear map from (Ck × Ck) to (Vk→Vk)). I collect here pairs of elements from Ck × (Vk→Vk) that we know belong to the quadratic form.

We name Clifford numbers as multinomials generated by scalars (from the field of the vector space of the Clifford algebra) and the elements γi of a fixed basis for that vector space.

<1, identity matrix>
i, identity matrix with −1 in ith diagonal position>
iγj, identity matrix with −1 in ith and jth diagonal positions>
and similarly with any product of a subset of the basis vectors.
<cos θ + (sin θ)γiγj, identity matrix except Oii = Ojj = cos 2θ and Oij = −Oji = sin 2θ>
If i, j, k and l are all distinct then
<(cos θ + (sin θ)γiγj)(cos φ + (sin φ)γkγl), identity matrix except Oii = Ojj = cos 2θ and Oij = −Oji = sin 2θ, Okk = Oll = cos 2φ and Oij = −Oji = sin 2φ>