This can be considered as a continuation of my last useful question:

Constructing groups of Type E7 with certain Tits Index

It is known that a quadratic form $q$ of dimension $12$, having splitting pattern $(2,4)$ (Vishik notation), gives an algebraic group with Tits index $E^{66} _{7,1}$ (anisotropic kernel is of type $D_6$) and trivial Tits algebra, by a Galois cohomological construction (see my old question for that.)

It is known that there are also groups with such Tits index, having Tits algebras with index $2$ or $4$.

For quite some time, i was under the impression that these are realized in the same manner, by taking some dimension $12$ quadratic form $q\in I^2$ (since we want none trivial Tits algebras), check back with an appropriate splitting pattern (all splitting patterns for dimension $12$ forms are classified) and we are done. I never really checked the details.

Now i did and it does not look like things work that way this time. /#/

Question: Can you give a method for constructing algebraic groups with Tits index $E^{66}_{7,1}$ and Tits algebras with index $2$ or $4$ ?

/#/ The index $2$ case might be easier. Writing 6 = n = 2*r and d = 2 (these are the parameters from Tits famous classification of indexes, meaning there are $r=3$ orbits) one can see that there should be three "splitting steps", giving the sequence $D_6, D_4$ x $A_1, A_1$ x $A_1$ x $A_1, \emptyset$ of anisotropic kernels. This would correspond to the splitting pattern $(1,2,3)$, which does not occur.