F4 (mathematics)

The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2.

This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin.

The fundamental representations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the Dynkin diagram in the order such that the double arrow points from the second to the third).

Embeddings of the maximal subgroups of F4 up to dimension 273 with associated projection matrix are shown below.

The 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F 4 in this Coxeter plane projection
Hasse diagram of F 4 root poset with edge labels identifying added simple root position