Can anyone provide an example of a (complex) representation which is simultaneously of real (with a structure map $j^2 = 1$)and quaternionic (with a structure map $j^2 = -1$) type? As this cannot be true for irreducible representations, we know it should be reducible.
Reference: Section 6, Chapter II, Representations of Compact Lie Groups by Brocker and Dieck.
Take any rep $V$ over $\Bbb R$ and consider $\Bbb H\otimes_{\Bbb R}V$ as a complex vector space.
Say $V$ is a representation of $G$ over $\Bbb R$. Assume we've picked a basis. Then $G$ acts by real matrices, and if we extend scalars (by tensoring with a larger ring of scalars) $G$ still acts by those real matrices.
Since $\Bbb C\subset\Bbb H$ we can interpret $\Bbb H\otimes_{\Bbb R}V$ as a complex vector space. The map $j$ comes from the scalar action of ${\bf j}\in\Bbb H$. Since $\Bbb H=\Bbb C\oplus\Bbb C{\bf j}$ this space is $\Bbb C\otimes_{\Bbb R}(V\oplus{\bf j}V)$ so it's of real type.