TITLE: Type of 26-dimensional representation of different real forms of the complex simple Lie algebra $F_4$
QUESTION [12 upvotes]: The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory.  In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.
Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms.  The three real forms are the compact real form, the split real form and a third form.  They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \operatorname{Tr} \operatorname{ad}_X \operatorname{ad}_Y$ has signature $(p,q)$, its index is $p-q$.  I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$.  The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.
I would like to know the following (pointers to the literature would also be greatly appreciated):
Questions

What is the type of $V$ under the different real forms?  I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.
And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!
Edit
Based on Jim's answer below, the representations are of real type in all cases.  From Bruce's answer, it would seem that for the split case $F_4^4$ the signature is (14,12).
In a rather convoluted calculation, I seem to find that for $F_4^{-20}$ the signature is (16,10), but I would like confirmation since I have seen at least one claim in the Physics literature (see last equation in §4) that it is (25,1).

REPLY [4 votes]: 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 $J(1,2,\mathbb{O}) = \{X\in \mathrm{M}(3,\mathbb{O}\otimes_\mathbb{R}\mathbb{C}) \, |\, I_1 \overline{X}^tI_1 = X \}$ 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.