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

ℝP²$\qquad\qquad\\;$ ℂP²$\qquad\qquad\\;$ ℍP²$\qquad\qquad\\;$ ↀP²

ℂP²$\qquad$ (ℂ⊗ℂ)P²$\qquad$ (ℂ⊗ℍ)P²$\qquad$ (ℂ⊗ↀ)P²

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

ↀP²$\quad\\;\\;$ (ↀ⊗ℂ)P²$\quad\\;\\;$ (ↀ⊗ℍ)P²$\quad\\;\\;$ (ↀ⊗ↀ)P²

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