# 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)?