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The compact group $F_4$ is the group of isometries of the octonionic projective plane $\mathbb{OP}^2$ endowed with an analog of Fubini-Study metric. I suspect the other real groups of type $F_4$ are the isometries groups of the octonionic hyperbolic plane and of the analogous objects build built from split octonions. (Related question on mathoverflow.) One of the noncompact real forms of $E_6$ is the group of projective transformations (collineations) of $\mathbb{OP}^2$. The groups of type $F_4$ and $E_6$ arise in this context because of their close relationship to the exceptional Jordan algebras of hermitian three by three matrices over octonions. Indeed -- the group $E_6$ preserves the determinant of these matrices and $F_4$ preserves the determinant and the trace.

The most geometric approach to the exceptional groups that I am aware of (and which goes in this direction) is that of Rosenfeld, but unfortunately . Unfortunately I don't have that book. He interprets groups of type $E_7$ and $E_8$ in a similar manner for $(\mathbb{C}\otimes\mathbb{O} ) \mathbb{P}^2$ and $(\mathbb{H}\otimes\mathbb{O} ) \mathbb{P}^2$. Some details and introduction to the subject is in Baez.

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The compact group $F_4$ is the group of isometries of the octonionic projective plane $\mathbb{OP}^2$ endowed with an analog of Fubini-Study metric. I suspect the other real groups of type $F_4$ are the isometries groups of the octonionic hyperbolic plane and of the analogous objects build from split octonions. (Related question on mathoverflow.) One of the noncompact real forms of $E_6$ is the group of projective transformations (collineations) of $\mathbb{OP}^2$. The groups of type $F_4$ and $E_6$ arise in this context because of their close relationship to the exceptional Jordan algebras of hermitian three by three matrices over octonions. Indeed -- the group $E_6$ preserves the determinant of these matrices and $F_4$ preserves the determinant and the trace.

The most geometric approach to the exceptional groups that I am aware of (and which goes in this direction) is that of Rosenfeld, but unfortunately I don't have that book. He interprets groups of type $E_7$ and $E_8$ in a similar manner for $(\mathbb{C}\otimes\mathbb{O} ) \mathbb{P}^2$ and $(\mathbb{H}\otimes\mathbb{O} ) \mathbb{P}^2$. Some details and introduction to the subject is in Baez.
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