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 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F4 in this Coxeter plane projection
The 48 root vectors of F4 can be found as the vertices of the 24-cell in two dual configurations, representing the vertices of a disphenoidal 288-cell if the edge lengths of the 24-cells are equal:
24-cell vertices:
24 roots by (±1, ±1, 0, 0), permuting coordinate positions
Dual 24-cell vertices:
8 roots by (±1, 0, 0, 0), permuting coordinate positions
Just as O(n) is the group of automorphisms which keep the quadratic polynomials x2 + y2 + ... invariant, F4 is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables).
Where x, y, z are real-valued and X, Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M2) and Tr(M3) of the hermitianoctonionmatrix:
The set of polynomials defines a 24-dimensional compact surface.
The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121738 in the OEIS):
The 52-dimensional representation is the adjoint representation, and the 26-dimensional one is the trace-free part of the action of F4 on the exceptional Albert algebra of dimension 27.
There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. 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.