3 expanded the answer to make it more understandable

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 (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 $J(1,2,\mathbb{O}) = \{X\in \mathrm{M}(3,\mathbb{O}\otimes_\mathbb{R}\mathbb{C}\, mathrm{M}(3,\mathbb{O}\otimes_\mathbb{R}\mathbb{C}) \, |\, I_1 \overline{X}^tI_1 = X \}$ where $I_1 = \mathrm{diag}(-1,1,1)$. Since Now the invariant form is given by computation of the signature of $A\mapsto \mathrm{Tr}(A^2)$ restricted to the space is a matter of trace-free matrices the result follows quite easilysimple 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.

The group $F_4^{-20}$ is according to Yokota 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 $I_1 = \mathrm{diag}(-1,1,1)$. Since the invariant form is given by $A\mapsto \mathrm{Tr}(A^2)$ restricted to the space of trace-free matrices the result follows quite easily.
The other two cases $F_4^{-52}$, $F_4^4$ follow similarly since they are the automorphism groups of $J(3,\mathbb{O}) = \{ X\in M(3,\mathbb{O}\,|\M(3,\mathbb{O})\,|\, X^t =X \}$ and $J(3,\mathbb{O}) = \{ X\in M(3,\mathbb{O}'\,|\M(3,\mathbb{O}')\,|\, X^t=X \}$ respectively.
The group $F_4^{-20}$ is according to Yokota 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 $I_1 = \mathrm{diag}(-1,1,1)$. Since the invariant form is given by $A\mapsto \mathrm{Tr}(A^2)$ restricted to the space of trace-free matrices the result follows quite easily.
The other two 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.