F4 lattice

In mathematics, F₄ is the name of a Lie group and also its Lie algebra f₄. It is one of the five exceptional simple Lie groups. F₄ has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.

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.

There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.

The F₄ Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E₈.

In older books and papers, F₄ is sometimes denoted by E₄.

The Dynkin diagram for F₄ is .

Its Weyl/Coxeter group ${\displaystyle G=W\left(F_{4}\right)}$ is the symmetry group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree ${\displaystyle \mu (G)=24}$ which is realized by the action on the 24-cell.

...
Wikipedia