All the forms of $F_4$ can be defined as automorphism groups of some Jordan algebra of three by three matrices with entries in octonions / split octonions / complexified octonions.  These algebras are all of dimension 27 over the appropriate field and the subspaces of trace-free matrices are the irreducible 26-dimensional representations of the various forms of $F_4$. The invariant quadratic form is $A\mapsto \mathrm{Tr}(A^2)$. (And the invariant cubic form is $A\mapsto \mathrm{det}(A)$.)

The group $F_4^{-20}$ is according to [Yokota][1] (but I guess that one can dig this up also out of the work of Veldkamp or Springer) the automorphism group of the real Jordan algebra $J(1,2,\mathbb{O}) = \{X\in \mathrm{M}(3,\mathbb{O}\otimes_\mathbb{R}\mathbb{C}) \\, |\\, I_1 \overline{X}^tI_1 = X \}$ where $I_1 = \mathrm{diag}(-1,1,1)$. Now the computation of the signature of $A\mapsto \mathrm{Tr}(A^2)$ is a matter of simple calculation.

The other two real cases $F_4^{-52}$, $F_4^4$ follow similarly since they are the automorphism groups of  $J(3,\mathbb{O}) = \{ X\in M(3,\mathbb{O})\\,|\\, X^t =X  \}$ and  $J(3,\mathbb{O}) = \{ X\in M(3,\mathbb{O}')\\,|\\,  X^t=X \}$ respectively.

  [1]: http://arxiv.org/abs/0902.0431