Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:


1) it's not isomorphic to its dual (in which case we call it 'complex')


2) it has a nondegenerate symmetric bilinear form (in which case we call it 'real')


3) it has a nondegenerate antisymmetric bilinear form (in which case we call it 'quaternionic')


It's 'real' in this sense iff it's the complexification of a representation on a real vector space, and it's 'quaternionic' in this sense iff it's the underlying complex representation of a representation on a quaternionic vector space.


Offhand, I know just **four compact Lie groups whose continuous irreducible representations on complex vector spaces are all either real or quaternionic in the above sense**:


1) the group Z/2 


2) the trivial group


3) the group SU(2)


4) the group SO(3)


Note that I'm so desperate for examples that I'm including 0-dimensional compact Lie groups, i.e. finite groups! 


1) is the group of unit-norm real numbers, 2) is a group covered by that, 3) is the group of unit-norm quaternions, and 4) is a group covered by *that*. This probably explains why these are all the examples I know. For 1), 2) and 4), all the continuous irreducible representations are in fact real.


**What are all the examples?** 



Which groups have only real and quaternionic irreducible representations?
An irreducible representation is real or quaternionic precisely when its
character is real-valued. By the Peter-Weyl theorem all characters are
real-valued precisely when every element in the group is conjugate to its
inverse. When the group is connected a more precise answer is as follows: The
Weyl group (in its tautological representation) must contain multiplication by
$-1$ and this is true precisely when all indecomposable root system factors have
that property. I don't remember off hand which indecomposable root systems have
this property but it is of course well known (type A is out, type B/C is in,
type D depends on the parity of the rank).


**Addendum**: Found the relevant places in Bourbaki. All characters are real-valued precisely when the element he calls $w\_0$ is $-1$ (Ch. VIII,Prop. 7.5.11) and one can also read off if a given representation is real or quaternionic (loc. cit. Prop 12). From the tables in Chapter 6 one gets that $w\_0=-1$ precisely for $A\_1$, B/C, D for even rank, $E\_7$, $E\_8$, $F\_4$ and $G\_2$.


