Consider a unitary real representation of a Lie group $G$ over a real Hilbert space $\mathcal{H}_\mathbb{R}$ \begin{equation} \rho:G\rightarrow U(\mathcal{H}_{\mathbb{R}}) \end{equation}

Taking the complexification $\mathcal{H}_{\mathbb{C}}:=\mathbb{C}\otimes_{\mathbb{R}},\mathcal{H}_{\mathbb{R}}$ it's straightforward how to extend $\rho$ to a complex unitary representation of $G$ over $\mathcal{H}_{\mathbb{C}}$.

Under which hypotheses is the irreducibility of $\rho$ preserved moving to the complexification?