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There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory.

For a Lie group G one can choose coordinates at the unit and expand the multiplication map to a power series which also gives a (possibly higher) formal group law.

What is know about the two realization problems and their relation? Which group laws come from complex oriented cohomology theories? Which from Lie groups? Is it sensible to associate Lie groups to cohomology theories or vice versa because of their group laws?

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Completions of Lie groups at the unit element will give you formal groups defined over the reals. If you want to compare those with the formal groups obtained from cohomology theories, you'll probably want to look for one-dimensional factors, but that's not going to lead to anything interesting since all one-dimensional formal groups over R are isomorphic to the additive group by the logarithm map.

On the other hand, looking at group schemes is much more promising. Elliptic curves give you elliptic cohomology, and one-dimensional summands of completions of higher abelian varieties have been studied by e.g. Ravenel.

As for your realization problem: I don't know about Lie groups, but for cohomology theories, the Landweber exact functor theorem gives you a partial, but still quite satisfying result.

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