Ag.algebraic geometry – Constructing algebraic groups of type E6 with algebras of divided tits

Let's assume our base camp. $ k $ It has zero characteristics.

From a series of articles by Borel and Siebenthal, it is known that there is an incrustation of root systems.

$ A_2 times A_2 times A_2 $ inside $ E_6 $.

This gives you a map

$ H ^ 1 (k, A_2 times A_2 times A_2) rightarrow H ^ 1 (k, E_6) $.

Consider the map

$ A_2 rightarrow A_2 times A_2 times A_2 $, that sends the CSA $ D $ to $ (D, D, D) $.

If we apply Galois cohomolgy again and compose this, we get

$ H ^ 1 (k, A_2) rightarrow H ^ 1 (k, E_6) $.

I assume that the algebra of tits of each group constructed in this way, is

$ D otimes D otimes D in H ^ 2 (k, mu_3) $ and therefore trivial, like the period of
$ D $ is $ 1 $ or $ 3 $.

It is also known, that each type group. $ E_6 $ (considered "mod 3"), which has trivial Tits algebras is anisotropic or divided.

Questions:

  1. My attempt to build a $ E_6 $ make sense?
  2. Is it known in the literature?
  3. It is the resulting anisotropic group iff. $ D $ has index $ 3 $ (I mean, yes $ A_2 $ is it anisotropic)?