Let $h_1\subset gl(V_1)$ and $h_2\subset gl(V_2)$ be two irreducible representations of reductive real Lie algebras.

When the representation of $h_1\oplus h_2$ in $V_1\otimes V_2$ is not irreducible?

I guess that the both representations $h_1\subset gl(V_1)$ and $h_2\subset gl(V_2)$ must be complex, i.e. on $V_i$ exists a complex structure commuting with $h_i$. But I can prove only that at least one of these representation is complex (otherwise the complisifications $h_1^\mathbb{C}\subset gl(V_1^\mathbb{C})$ and $h_2^\mathbb{C}\subset gl(V_2^\mathbb{C})$ are irreducible, and consequently $h_1^\mathbb{C}\oplus h_2^\mathbb{C}$ is irreducible in $V_1^\mathbb{C}\otimes_\mathbb{C} V_2^\mathbb{C}$, which gives a contradiction).