Characterising the bounded derived category of an hereditary abelian category

We show that if a (not necessarily algebraic) triangulated category T contains an admissible hereditary abelian subcategory H, then we can lift the inclusion of H into T to a fully faithful triangle functor from the whole of the bounded derived category of H to T. This allows us prove, for example, that a triangulated category T is triangle equivalent to the bounded derived category of an hereditary abelian category if and only if T admits a split, bounded t-structure.