We discuss holomorphic induction of unitary representations of a class of infinite-dimensional Lie groups, specifically regular BCH (Baker–Campbell–Hausdorff) Fréchet-Lie groups. After clarifying the relevance and usual approach to holomorphic induction, we explain how the theory can be extended beyond the very restrictive setting where the representation being induced is required to be norm-continuous, whilst generalizing important representation-theoretic results. We also explain how this relates to representations satisfying a so-called positive energy condition.