On complex H-type Lie algebras

H-type Lie algebras were introduced by Kaplan as a class of real Lie algebras generalizing the familiar Heisenberg Lie algebra $\mathfrak{h}^3$. The H-type property depends on a choice of inner product on the Lie algebra $\mathfrak{g}$. Among the H-type Lie algebras are the complex Heisenberg Lie algebras $\mathfrak{h}^{2n+1}_{\mathbb{C}}$, for which the standard Euclidean inner product not only satisfies the H-type condition, but is also compatible with the complex structure, in that it is Hermitian. We show that, up to isometric isomorphism, these are the only complex Lie algebras with an inner product satisfying both conditions. In other words, the family $\mathfrak{h}^{2n+1}_{\mathbb{C}}$ comprises all of the complex H-type Lie algebras.