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](http://arxiv.org/abs/hep-th/0301050v1) (see last equation in §4) that it is (25,1).



Type of 26-dimensional representation of different real forms of the complex simple Lie algebra $F_4$
I think the best way to see the signature of these quadratic forms is by using the formula from "A Classification Theorem for Albert Algebras" by R. Parimala, R. Sridharan, and Maneesh L. Thakur, Trans. AMS 350 #3, March 1998.


All forms of $F\_4$ arise from Albert algebras. Over $R$, these are 27-dimensional algebras, whose unital automorphisms form groups of type $F\_4$. They are classified, over fields of characteristic neither $2$ nor $3$, by cohomological invariants $f\_3$ and $f\_5$. These cohomological invariants determine 3-fold and 5-fold Pfister forms, $\phi\_3$ and $\phi\_5$ respectively.


The formula of P-S-T (above), or maybe originally due to Serre, is that for an Albert algebra $A$ over $k$,
$$Q\_A \perp \phi\_3 \cong <2,2,2> \perp \phi\_5.$$


Now there are only two Pfister forms over $R$ for $\phi\_3$ and $\phi\_5$. The signature of $\phi\_3$ is either $(8,0)$ or $(4,4)$. Similarly, the signature of $\phi\_5$ is either $(32,0)$ or $(16,16)$. The signature of $<2,2,2>$ is $(3,0)$. Hence the possibilities for the signature $(p,n)$ of $Q\_A$ are:
$$(p,n) + (8,0) = (3,0) + (32,0),$$
$$(p,n) + (8,0) = (3,0) + (16,16),$$
$$(p,n) + (4,4) = (3,0) + (32,0),$$
$$(p,n) + (4,4) = (3,0) + (16,16).$$


Only three cases are possible: $(p,n) = (27,0)$ or $(p,n) = (11,16)$ or $(p,n) = (15,12)$.


As $F\_4$ acts on the orthogonal complement of the identity, and the identity has positive norm, the possible signatures for the 26-dimensional rep of $F\_4$ are:
$$(26,0), (10,16), (14,12).$$


