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 where all continuous irreducible representations are either real or quaternionic: 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! Note that 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 <i>that</i>. That probably explains why these are all the examples I know. For 2) and 4), all the continuous irreducible representations are in fact real. <b>What are all the examples?</b>