Let $\mathbb{H}$ be the four-dimensional real vector space of quaternions, G multiplicative group $\mathbb{H}\backslash\{0\}$ and H multiplicative group $\mathbb{C}\backslash \{0\}$. Let $\pi$ be a representation of group G on the 4-dimensional real vector space $\mathbb{H}$ defined by $\pi(\alpha)\beta=\alpha\beta,\quad\alpha\in G,\,\beta\in\mathbb{H}$ and $\rho$ analogously defined representation of group H on the 2-dimensional real vector space $\mathbb{C}$.
This should be an example of a case when representations $\pi$ and $\rho$ are irreducible, but their outer tensor product $\pi\times\rho$ is not. I am trying to find a $(\pi\times\rho)$-invariant subspace of $\mathbb{H}\otimes \mathbb{C}$ that shows that $\pi\times\rho$ is reducible, but... I can't. Any hints?