I am trying to construct arithmetic subgroups in the $F_4^{(-20)}$ Lie group. I am aware that algebraic groups of type $F_4$ classify Albert algebras. To construct $\mathbb{Q}-$forms I need the following information:
Why is an Albert Algebra classified completely by cohomological invariants? What are the exact descriptions of the mod 2 and mod 3 invariants and how do they work over localizations $\mathbb{Q}_p$?
What is the kernel and image of the map on Galois cohomology $$H^1(\mathbb{Q}, F_4) \longrightarrow \Pi_p \; H^1(\mathbb{Q}_p, F_4)?$$
What Albert algebras $A$ over $\mathbb{R}$ have automorphism groups $F_4^{(-20)}$? Is there some version of Witt index?
I want to know if there is a "cyclic construction" for Albert algebras, where, given a list of cohomological invariants at every prime that satisfy some compatibility criterion to come from a global Albert algebra, one can construct the global algebra explicitly.
I am having trouble finding a reference that even so much as discusses Albert algebras over local fields, much less the Hasse principle for them. Please share any references that are relevant.