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 $\kappa(X,Y) = \mathrm{Tr} \mathrm{ad}_X \mathrm{ad}_Y$ 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!