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Freudenthal's magic square of Lie algebras
and the corresponding square of projective planes:

ℝP²            ℂP²           ℍP²           ↀP²

ℂP²       (ℂ⊗ℂ)P²     (ℂ⊗ℍ)P²     (ℂ⊗ↀ)P²

ℍP²       (ℍ⊗ℂ)P²    (ℍ⊗ℍ)P²$    (ℍ⊗ↀ)P²

ↀP²     (ↀ⊗ℂ)P²    (ↀ⊗ℍ)P²    (ↀ⊗ↀ)P²

Also: have a look at John Baez's cheat sheet,
and at the subsections G₂, F₄, E₆, E₇, E₈.